Download Lesson #74 - Investigating the Unit Circle Common Core Algebra 2

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Lesson #74 - Investigating the Unit Circle
Common Core Algebra 2
Circular Trigonometry is at its essence a study of patterns. We will be exploring some of those patterns in the
unit circle which is really the unifying concept in the study of this area of mathematics.
Exercise #1: The unit circle is the parent function for all circles. It is the most basic circle which can be dilated
and translated to become any other circle on the coordinate plane.
1. Based on the graph, what is the center of the unit circle?
2. Based on the graph, what is the radius of the unit circle?
(1,0)
3. Based on the graph, what is the equation of the unit circle?
4. There are four points in the unit circle that should be very obvious to
find. One is already labelled. Label the other three “easy” points on the
unit circle.
Exercise #2: In the unit circle diagram to the right, 16 points are shown on
the circle.
a) Label the point (1,0), A. Going in a counterclockwise direction around
the circle, label the other given points B-P.
b) Label the four quadrants on the graph.
c) Quadrant I:
i.
What are the signs of the x-coordinates and the y-coordinates in
quadrant I?
ii.
Of the three points in quadrant I, which has the largest x-value?
iii.
Of the three points in quadrant I, which has the largest y-value?
iv.
Of the three points in quadrant I, which has the smallest x-value?
v.
Of the three points in quadrant I, which has the smallest y-value?
vi.
Of the three points in quadrant I, which has an x-value and a y-value that are about the same?
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d) Quadrant II (Label the points A-P as you did on the other page).
i.
What are the signs of the x-coordinates and the y-coordinates in
quadrant II?
ii.
Of the three points in quadrant II which has the largest absolute
value for x (for example, -.9 has a larger absolute value than -.5)?
iii.
Of the three points in quadrant II, which has the largest absolute
value for y?
iv.
Of the three points in quadrant II, which has the smallest
absolute value for x?
v.
Of the three points in quadrant II, which has the smallest absolute value for y?
vi.
Of the three points in quadrant II, which has an x-value and a y-value with the same absolute value for x
and y?
e) Quadrant III
i.
What are the signs of the x-coordinates and the y-coordinates in quadrant III?
ii.
Of the three points in quadrant III which has the largest absolute value for x?
iii.
Of the three points in quadrant III, which has the largest absolute value for y?
iv.
Of the three points in quadrant III, which has the smallest absolute value for x?
v.
Of the three points in quadrant III, which has the smallest absolute value for y?
vi.
Of the three points in quadrant III, which has an x-value and a y-value with the same absolute value for
x and y?
f) Quadrant IV
i.
What are the signs of the x-coordinates and the y-coordinates in quadrant IV?
ii.
Of the three points in quadrant IV which has the largest absolute value for x?
iii.
Of the three points in quadrant IV, which has the largest absolute value for y?
iv.
Of the three points in quadrant IV, which has the smallest absolute value for x?
v.
Of the three points in quadrant IV, which has the smallest absolute value for y?
vi.
Of the three points in quadrant IV, which has an x-value and a y-value with the same absolute value for
x and y?
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Exercise #3: Standard Position for an Angle and Quadrantal Angles
a) Standard Position for an angle: An angle with its initial (starting) side on the
positive x-axis and its vertex on the origin. The other side of the angle
is called the terminal (ending) side.
**Positive angles go __________________ while negative angles go
_______________.
How many degrees are in one rotation?
b) Quadrantal Angle: An angle with its terminal side on an axes.
On the axes to the right, label the quadrantal angles with their degree measures.
Exercise #4: Angle Measures in Degrees
(Label the points A-P on the unit circle as you did on the other page).
Each of the points, A-P, on the unit circle is a special point you will be
working with later in the unit. For now, we will learn the angle measures, in
degrees, of an angle in “standard position” that passes through each of these
points.
a) Draw an angle in standard position through each of the points.
b) Label the angle measure, in degrees, of each quadrantal angle.
c) The four angles that pass through the points that are closest to the x-axis
are exactly 30° away from the x-axis. Using this information, label these four angles (These angles pass
through B, H, J, and P).
d) The four angles that pass through the points that are midway between the x-axis and y-axis are 45° away
from the x-axis. Using this information, label these four angles.
e) The remaining four angles that are furthest from the x-axis are exactly 60° away from the x-axis. Using
this information, label these four angles.
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Exercise #5: The Circumference of the Unit Circle
To the right is the unit circle. This diagram is slightly larger than
the others, to help with the next activity.
a) The radius of the unit circle is _____.
b) Take a string and measure the radius of the circle. Make a
mark on the string with a pen for the length of the radius
which is ____.
c) Starting from the (1,0) and working counterclockwise
around the unit circle, wrap the string around the unit circle
and mark off one unit length on the circle.
d) Move the beginning of the string to that point and continue
measuring unit lengths around the unit circle.
e) What is the circumference?
f) From your study in geometry, what is the formula for the circumference of a circle? Use the formula that contains the
radius, not the diameter.
g) Calculate the circumference of the unit circle. Is this answer approximately the same your answer to (e)?
Exercise #6: Last year you worked with converting between degree
measure and radian measure for angles. In this unit, we will look at
another way to define radian measure.
The radian measure of an angle is the length of the arc on the unit circle
subtended by the angle in standard position.
Therefore, the point above where you marked the first unit, corresponds to
an angle of 1 radian.
a) How many radians are in the full rotation? (See your answer to
3g) Label this value on the unit circle to the right where you
would label 360°.
b) How many radians are in half a rotation?
c) Label this value on the unit circle to the right where you would
label 180° .
d) Use this information to label the other two quadrantal angles.
e) Assume the diagonal lines represent angles that are 45° away from the x-axis. Label these angle measures in
radians.
Note: All of the exercises in this lesson should make intuitive sense the first time you see them EXCEPT for Exercise #6.
This usually takes time to develop understanding.
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Investigating the Unit Circle
Common Core Algebra 2 Homework
1. Explain why the equation of the unit circle is x 2 + y 2 = 1 .
2. The following are the points on the basic unit circle diagram,
rounded to the nearest hundredth where necessary. Label each
point in the diagram.
Hint: If you are having trouble, use the positive and negative values
of x and y in each quadrant and the relationship between x and y
(larger/smaller/same) to determine where the points should go.
(0, 1)
(.71, .71)
(-.87, -.5)
(0, -1)
(-.71, -.71)
(1,0)
(.5, -.87)
(-.5, -.87)
(.87, -.5)
(-.87,.5)
(.71, -.71)
(-1,0)
(.5, .87)
(-.71, .71)
(.87, .5)
(-.5, .87)
3. As you can see, none of the x or y values of the points on the unit circle is greater than 1. Explain why this has to be
true.
4. Label each of the angle measures on the unit circle diagram in degrees. If possible, see if you can do so without using
exercise #4 from the lesson.
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5. Label each the angle measures in the unit circle diagram below in radians. If possible, do so without using exercise
#6 from the lesson.
6. Fill in the following chart with equivalent angle measures in radians and degrees. Use question 4 and 5 if you need
help.
Degrees
Radians
0
45
90
225
3
p
4
p
270
7
p
4
2p
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Lesson #75 – The Three Basic Trigonometric Functions and Their Reciprocals
Common Core Algebra 2
The table below reviews the Pythagorean Theorem and the three basic trigonometric functions.
Term
Pythagorean Theorem:
Sine of an angle
Cosine of an angle
Tangent of an angle
Formula/Symbol
Important Information
a 2  b2  c 2
This can only be used with right triangles!
opposite
hypotenuse
adjacent
cos( A) =
hypotenuse
opposite
tan( A) =
adjacent
In a right triangle, the ratio between the side Opposite an angle
and the Hypotenuse
sin( A) =
In a right triangle, the ratio between the side Adjacent or next to
an angle and the Hypotenuse
In a right triangle, the ratio between the side Opposite an angle
and the side Adjacent to the angle
Memory Device:
Note: There is a lot more to the definitions of sine, cosine, and tangent. We will be expanding our understanding of
these trig. functions throughout the unit.
Exercise #1: Find the trigonometric function(s) with the given information.
a)
In right triangle DEF, E is the right angle. If DE=1 and
DF=3, find sinF, cosF, and tanF.
F
D
E
b)
In right triangle XYZ, Z is the right angle. If XY=10, YZ=
X
5 3 , and XZ=5, find the following trigonometric values.
SinX =
SinY =
CosX =
CosY =
TanX =
TanY=
Y
Z
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Exercise #2: If the value of sine, cosine, or tangent of an angle is given, it is possible to find the values of the other two
trigonometric functions.
a) If sin x 
4
, find the values of the other two trigonometric functions.
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Steps
1. Draw a right triangle and label
the sides with the ratio
you are given. (Remember, the size
of the triangle does not matter.)
2. Find the third side using
Pythagorean theorem.
3. Find the other basic trig ratios
using SOH-CAH-TOA.
b) If t is an acute angle and tan t 
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, find the values of the other two trigonometric functions. Express your
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answers in simplest radical form.
Exercise #3: There are three ratios that are also trigonometric functions. They are called the reciprocal trigonometric
functions because they are reciprocals of the three original trigonometric functions. Each of the reciprocal functions also
has its own name.
Name
Abbreviation
Ratio with sides of
a right triangle
Relation to sine,
cosine, or tangent
Value with respect
to <A
A
Cosecant
Secant
13
12
B
Cotangent
C
5
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Life is not perfect. Sine and cosecant are reciprocals and cosine and secant are reciprocals. In this case, opposites
attract.
Exercise #4: Find the values of the 6 trigonometric functions for A and B.
.
sinA=
sinB=
cosA=
cosB=
tanA=
tanB=
cscA=
cscB=
secA=
secB=
cotA=
cotB=
B
20
29
A
21
C
Exercise #5: Similar to exercise #2, any one trigonometric function value can be used
to find the other five trigonometric values.
a) If sin x =
4
, find the values of the other five trigonometric functions.
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We learned that
csc 
Therefore, if sin q =
4
,
5
cscq =
If we know a basic trig
ratio, we can flip that value
to find the reciprocal ratio.
b) If
q is an acute angle and secq = 3, find the value of tanq in simplest radical form.
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Exercise #6: Use the questions below to find the radian measures for the following angles.
a) From the previous lesson, 90° is equivalent to what radian measure?
b) We saw that 45° is equivalent to ______ radians because 45° is half of 90° and
1
1
p is half of p .
4
2
c) 30° is ______ of 90° . Use this fact to find the radian measure that is equivalent to 30° .
d) 60° is ______ of 90° . Use this fact to find the radian measure that is equivalent to 60° .
e) Label these radian measures on the portion of the unit circle to the
right. Even though these first quadrant radian measures should make
sense, you will want to memorize them as soon as possible because it
will make future calculations easier.
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The Three Basic Trigonometric Functions and Their Reciprocals
Common Core Algebra 2 Homework
1.
In right triangle ABC, C is the right angle. If AB=10 and BC=8,
find sinA and sinB.
A
B
C
2.
In right triangle DEF, E is the right angle. If EF=14 and DF=19,
find tanF.
F
E
3.
In right triangle XYZ, Z is the right angle. If XY=12, YZ= 8
and XZ=4, find the following trigonometric values.
sinX =
sinY =
cosX =
cosY =
tanX =
tanY=
secX =
secY =
cscX =
cscY =
cotX =
cotY=
2,
D
X
Y
Z
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4. If t is an acute angle and cos t 
5
, find the values of sint and tant .
13
15
, find sinθ. Express your answer in simplest radical form.
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5. If  is an acute angle and sin q =
6. If t is an acute angle and csct =
25
, find the values of the other five trigonometric functions.
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7. If t is an acute angle and cos A =
2
, find the values of the other five trigonometric functions. Express your answers
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in simplest radical form.
8. If β is an acute angle and cot  
3
, find the value of sin(β) in simplest radical form.
3
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9. Fill in the blanks on the unit circle
diagram with the appropriate degree and
radian measures.
10. Use the diagram from the previous question to fill in the chart below with the equivalent radian measures for each
degree measure.
Degrees
Radians
0
30
45
60
90
180
270
360
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Lesson #76 – Trigonometric Values for Special Angles
Common Core Algebra 2
The two special triangles you learned last year were _________ and _________ triangles. They are called “special”
because they have specific ratios between their sides. Since all 45-45-90 triangles are similar and all 30-60-90 triangles
are similar, size does not matter. The ratios are always true. These ratios are shown in the two diagrams below.
Label the equivalent radian measures of each angle.
30
45
x 2
2x
x 3
x
90
45
90
x
Special Angle Trigonometric Values Summary Chart
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x
Exact Values
30
45
Angles in
Degrees
Angles in
Radians
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Do your calculations for the exact values first on the next page.
Copy your final answers in into the table on the left. You must know
the bolded part of the table by memory.
The estimated values (shown below) for these trigonometric functions
can be found using your calculator.
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sin 
sin 
cos 
tan 
csc
sec
cot 
cos 
tan 
Estimated Values
30
45
0.5
0.7071
0.866 0.7071
0.577
1
2
1.414
1.155 1.414
1.732
1
60
0.866
0.5
1.732
1.155
2
0.577
csc
To verify that the estimated values are correct, type sin45° and
tan30° in your calculator. Make sure you are in degree mode.
sec
The estimated values can always be found this way. The calculator
can also be used to check exact values; however, the calculator will
never give the exact values for the special angles except those that are
rational numbers (.5, 1, or 2).
cot 
30
45
x 2
2x
x 3
x
90
45
x
90
60
x
Exercise #1: Use the two triangles to find the exact (in simplest radical form when necessary) trigonometric values for
sine, cosine, and tangent. Use the reciprocal relationships to find the values for secant, cosecant, and cotangent. All
answers should be in simplest radical form.
30
45
60
Pattern/Method

sin 
cos 
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tan 
csc
sec
cot 
Exercise #2: HINTS FOR MEMORIZATION: The 1-2-3, 3-2-1 Method
a) Look at the table on the first page of the lesson. What pattern do you notice in the sine values?
b) What pattern do you notice in the cosine values?
c) In the table to the right, find the sine and cosine values using
this method.
If you can memorize the sine and cosine values of the special angles,
you can use a couple of relationships to find the rest.
d) The tangent quotient identity: The sine value of an angle
divided by the cosine value of an angle is equal to the tangent
sin A
= tan A . Use SOHvalue of the angle. In other words,
cos A
Degree
Measures
Radian
Measures
sin 
p
p
p
6
4
3
cos 
tan 
CAH-TOA to show why this is true.
For the three reciprocal functions, just find the reciprocal of the appropriate function in the table above.
Exercise #3: Find the exact values of each of the trigonometric functions below. Before doing so, cover up the table
above and see if you can make it again on your own using the memory devices. You may also work from the right
triangles themselves.
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a)
cos(45°)=
d)
tan
p
4
=
g) csc(45°)=
j)
sec
p
4
=
b) tan(30°)=
e)
p
sin =
6
h) cot(60°)=
k)
p
cot =
4
c)
sin(60°)=
f)
cos =
3
i)
sec(30°)=
l)
csc =
6
p
p
Exercise #4: Where did you see the special angles earlier this unit?
Trigonometric Values for Special Angles
Common Core Algebra 2 Homework
1. Fill in the special angle values table from memory or using the special triangles. You may use the quotient
relationship for tangent.
Degree
Measures
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Radian
Measures
sin 
cos 
tan 
m) sin(45°)=
p
2. Find the exact values of each of the trigonometric
functions below. You may use the memory devices learned
in this unit or work from the right triangles themselves.
n) cos(30°)=
p
p)
cos
=
q)
tan
s)
cot(45°)=
t)
sec(60°)=
v)
sec
w)
csc =
3
4
p
6
=
6
p
=
o) tan(60°)=
r)
p
sin =
3
u) csc(30°)=
x)
p
cot =
3
The next lesson will start to connect the unit circle with the trigonometric functions. Answer the following
questions to review what you already know about the unit circle.
3. The equation of the unit circle is ____________ because the radius of the circle is ____ and the center of
the circle is _______.
4. Label all of the degree measures on the unit circle below. Do so from memory.
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5. On the same unit circle, label the radian measures that are equivalent to the following degree measures:
0°,30°,45°,60°,90°,135°,180°,225°,270°,315°,360° . You should have the radian values for the first
quadrant angles and the quadrantal angles memorized. As we started to do in the lesson, if there is a
1
p
fraction in the radian measure, write p in the fraction. For example, write p as
2
2
HOMEWORK CONTINUES ON THE NEXT PAGE
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6. On the unit circle below, label the points A-P starting
with the point (1,0) and moving counterclockwise
around the circle.
a. Write the coordinaes of the the four points on the quadrantal angles.
b. Compare points B and D. Which has a larger x-value? Which has a larger y-value?
c. Compare points C and G. What is the same about these two points? What is different about these two
points?
d. Complete the following directions with point C.
i. Draw a line from point C to the origin.
ii. Draw a vertical line from point C to the x-axis.
iii. You now have a triangle with its base on the x-axis. How long is the hypotenuse of the
triangle? (What is the radius of the unit circle?) Put this number on the hypotenuse of
the triangle.
iv. Label the base of the triangle x and the height of the triangle y to correspond with the
coordinates of point C.
v. Label the angle in the triangle that is next to the origin, q .
vi. Find the following trigonometric ratios using the three labeled sides.
sin q =
cos q =
tan q =
e.
Explain why the x-coordinate and the y-coordinate of a point on the unit circle can never
exceed 1.
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Lesson #77 – The Unit Circle Part 1
Common Core Algebra 2
Exercise #1: Finding the rest of the radian measures on the unit circle.
You know the radian measures for angles of 0°,30°,45°,60°,90°,135°,180°,225°,270°,315°, and 360° already.
Label them on the unit circle to the right.
Recall, to find 150°,210°, and 330° , add or
subtract 30° from 180° or 360° depending on
the quadrant.
a) Therefore, to find the radian measures that are
equivalent to 150°,210°, and 330° ,
add or subtract _____ from _____ or
_____ depending on the quadrant.
b) Label these angles on the unit circle diagram.
Recall, to find 120°,240°, and 300° , add or subtract 60° from 180° or 360° depending on the quadrant.
c) Therefore, to find the radian measures that are equivalent to 120°,240°, and 300° , add or subtract _____
from _____ or _____ depending on the quadrant.
d) Label these angles on the unit circle diagram.
Note: Even though we learned how to find the radian measures for 135°,225°, and 315° by thinking about fractions of
a circle, we could also add or subtract
p
4
from
p
or 2p depending on the quadrant.
These values that are added or subtracted to find the angles in the other quadrants are known as reference angles. The
reference angles we have worked with so far are ______, ______, and _____ in degrees and _______, _______, and
______ in radians.
Looking for patterns in the radian measures on the unit circle can make learning them much easier.
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Exercise #2: To the right is the unit circle.
a) Draw a 30° angle on the unit circle.
b) Draw a vertical line from the point where the unit circle intersects 30° to
the x-axis.
c) You now have a triangle with its base on the x-axis. How long is the
hypotenuse of the triangle?
d) Label the base of the triangle x and the height of the triangle y to
correspond with the coordinates of the point intersected by the 30° angle.
e) Find the following trigonometric ratios using the three labeled sides.
sin 30° =
cos 30° =
tan 30° =
Because the hypotenuse is 1, the sine of any angle, q , on the unit circle is
equal to the y-value of the point intersected by the angle.
Because the hypotenuse is 1, the cosine of any angle, q , on the unit circle is equal to the x-value of the point
intersected by the angle.
f) Therefore, the point on the unit circle intersected by an angle of 30° is (______ , _______).
g) Using the same reasoning, the point on the unit circle intersected by an angle of 45° is (______ , _______).
h) Using the same reasoning, the point on the unit circle intersected by an angle of 60° is (______ , _______).
i)
What point on the unit circle is intersected by an angle of 90° ? (______ , _______).
Working backwards, what is sin90° ? _______
Working backwards, what is cos90° ? _______
Exercise #3: Complete the unit circle plate activity. Depending on the time, use what you already did this lesson to
complete parts of the unit circle when you can.
Basic Circle Setup
(1)
(2)
(3)
(4)
Flatten out the plate as much as possible.
Draw a circle using the indent in the plate. This is the unit circle which has a radius of 1.
Fold the plate carefully in half, then into quarters.
Open the plate back up and, using one color, draw the lines made by the creases. Choose and label your x- axis and yaxis.
(5) Label each quadrant (I-IV). Put the sign values of the x-coordinate and y-coordinate in each quadrant.
Quadrantal Angles
(6) Label the quadrantal angles in degrees and radians.
(7) Label the points where the unit circle intersects the axes.
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Special Angles
(8) Use your triangles to draw angles of 30°, 45°, and 60° in the 1st quadrant (using different colors for each). Label the
angle measures in both degrees and radians.
(9) As you continue, the following should be true:
The same color is used for all of the quadrantal angles.
The same color is used for all of the angles which are 30° away from the x-axis.
The same color is used for all of the angles which are 45° away from the x-axis.
The same color is used for all of the angles which are 60° away from the x-axis.
Special Angles in other quadrants
(10)
Use a ruler to extend the lines for 30°, 45°, and 60° into Quadrant III (using same color as the each original
angle). Label these angles with their degree and radian measures based on their distance from 180° or p .
(11)
Use your triangles to draw angles that are 30°, 45°, and 60° above the negative x-axis (quadrant II). Label their
angle measures in degrees and radians based on their distance from 180° or p .
(12)
Extend the quadrant II lines into quadrant IV. label the angle measures based on their distance from 360° or or
.
2p
Points on the Unit Circle
(13)
Since the length of the radius of the unit circle is 1, the length of the hypotenuse of your triangles is also 1. Label
this on both sides of the two triangles.
(14)
Find the exact lengths of the two legs of your triangles using your special angle chart. Label these values on both
sides of your triangles.
(15)
Use these values to label the points where your angles intersect the circle by placing the triangles with 30°, 45°,
and 60° angles in each of the four spots where they are a reference angle. PUT THE BASE ON THE X-AXIS, NOT
THE Y-AXIS. Label the points using the same color as the color you used for the angles.
(16)
Look for patterns on your unit circle.
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The Unit Circle Part 1
Common Core Algebra 2 Homework
1. If you did not finish it in class, use the directions to complete the unit circle plate. Use a pencil so you can change
it if you have any errors from working on your own.
2. Fill in the unit circle diagram below. Use one color to do as much as you can do with without looking at the unit
circle plate. Use another color to do the rest. This will let you see what you understand/remember and what you
still need to learn.
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Lesson #78 – Triangles on the Coordinate Plane
Common Core Algebra 2
Recall from last lesson that the x-coordinate of a point on the unit circle intersected by an angle in standard position is the
cosine of the angle.
The y-coordinate of a point on the unit circle intersected by an angle in standard position is the sine of the angle.
Exercise #1: Look at the unit circle plate to find the following values:
a) cos120°
d) cos
b) sin300°
5p
4
c) sin
e) cos270°
f)
sin
5p
6
p
2
If trigonometric values only existed for angles that fit in right triangles, the problems above would be impossible. How
does it make sense to find the sin300°? Reference angles and reference triangles make it possible.
An angle of any size can be drawn in standard position on the coordinate plane.
From that angle, a reference triangle can be drawn to the closest x-axis.
Exercise #2: The following activity investigates the signs of the trigonometric functions for angles in each quadrant.
1. The sign of the adjacent side is determined by the sign of x in that quadrant.
2. The sign of the opposite side is determined by the sign of y in that quadrant.
3. The hypotenuse is always positive.
a) An angle in standard position is drawn in each quadrant.
b) Draw a reference triangle in each quadrant
c) Label each side of the triangle as positive (+) or negative (-) using the 3 rules above.
d) Use quadrant II as an example to complete the textboxes for quadrants I, III, and IV.
Quadrant I
Quadrant II
sin=
+h
cos=
tan=
+y
-x
sin= o     number
h 
cos= a     number
h 
tan= o     number
a 
sec=
sec= h    - number
a 
csc=
csc= h    + number
o 
cot=
cot= a     number
o 
26
Quadrant III
Quadrant IV
sin=
sin=
cos=
cos=
tan=
tan=
sec=
sec=
csc=
csc=
cot=
cot=
The following chart helps you remember which of the three basic
trigonometric functions are positive in each quadrant.
There is a mnemonic statement that may be helpful for remembering
the positive trig values (and their reciprocals) in each quadrant.
A S T C - All Students Take Calculus!
(We start the brainwashing early!)
Exercise #3: An angle,
5
6
q , in standard position terminates in the second quadrant. If the tanq = - , find sinq and
cosq .
Exercise #4: An angle,
q , in standard position terminates in the fourth quadrant. If the cosq =
12
, find cscq .
13
27
Exercise #5: An angle,
3
2
q , in standard position terminates in the fourth quadrant. If the cot q = - , find the values of
the other five trigonometric functions.
Exercise #6: Given a point on the terminal side of an angle, find the following trigonometric values.
a) The terminal side of an angle, q , in standard position passes through the
point, (-12,-35). Find cos and cot.
b) The terminal side of an angle,


q , in standard position passes through the
point, 5 3, 5 . Find sin and csc.
c) The terminal side of , an angle in standard position, passes through the
(
)
point, -5,1 . Find the values of the six trigonometric functions.
28
Triangles on the Coordinate Plane
Common Core Algebra 2 Homework
3
4
1. An angle,
q , in standard position terminates in the third quadrant. If the sinq = - , find tanq and cosq .
2. An angle,
q , in standard position terminates in the second quadrant. If the cosq = -
3. An angle,
q , in standard position terminates in the fourth quadrant. If the secq = , find the values of the other
9
, find cscq and cot q .
41
5
3
five trigonometric functions.
4. The terminal side of an angle, A, in standard position passes through the point, (-5,-12). Find cosA and cotA.
5. The terminal side of an angle in standard position passes through the point,
(-2 2,2 2 )
. Find sin and csc.
(
)
29
6. The terminal side of , an angle in standard position, passes through the point, 20,-6 . Find the values of the
six trigonometric functions.
7. Fill in the unit circle diagram below. Use one color to do as much as you can do with without looking at the unit
circle plate. Use another color to do the rest. This will let you see what you understand/remember and what you
still need to learn.
30
Lesson #79 – The Unit Circle Part 2
Common Core Algebra 2
How is the unit circle helpful?
a. It allows us to work with trigonometric functions without using a triangle because the hypotenuse is always
one. All we need is a point on the unit circle, (x,y).
b. This allows us to extend the trigonometric functions to any angle, not just those that fit in a right triangle.
c. The x-value of the point on the unit circle is the same as the cosine of the angle.
d. The y-value of the point on the unit circle is the same as the sine of the angle.
e. It allows us to find the trigonometric values of the quadrantal angles when we cannot draw a triangle (90°,
180°, etc.).
f. It is the basis for creating the function graphs of sine and cosine.
Exercise #1: The unit circle gives us a helpful identity involving sine and cosine.
a) Draw the unit circle on the axes to the right.
b) On the axes draw an angle with its terminal side in the first quadrant.
c) Draw a triangle using this angle similar to previous lessons.
d) Label the sides of the triangle.
e)
sin  
cos 
f) What is the equation of the unit circle?
g) Substitute the appropriate trigonometric functions for x and y in the
equation of the unit circle.
This equation is known as the Pythagorean Identity. It is called an identity
because it is true for any angle. The Pythagorean Identity can be used to find the
value of sine when given the value of cosine and visa versa. It is still probably
easier and safer to draw a triangle in the appropriate quadrant to solve these
problems to make sure the sign of the trigonometric function is correct.
Exercise #2: If sinq =
7
and q is an angle that terminates in quadrant II, find cosq .
8
a) Solve this problem by drawing a triangle in the appropriate quadrant.
b) Solve this problem using the Pythagorean Identity.
Another strength of the first method is that it is easier work with the other four trigonometric functions.
31
Working with the unit circle to solve problems.
When you are in a testing situation, you will not have the unit circle diagram with you. Still, the concepts on the diagram
and a quick sketch of the important part of the diagram can help you solve a number of problems.
What you need to have memorized
90°
180°
270°
360°
Degree
Measures
*Radian
Measures
p
p
3p
2
2p
sin 
Any point on the unit circle is (cosq ,sinq )
Degree
Measures
*Radian
Measures
2
cos 
*The rest of the radian measures can be found using reference angles.
30°
45°
60°
p
p
p
6
1
2
4
3
2
2
3
2
2
2
3
2
1
2
Each angle on the unit circle has a degree measure, a radian measure, and a point where it intersects the circle. One piece
of information can be used to find the other one or both of the other two.
Exercise #4: Answer the following questions involving points or angles on the unit circle.
NOTE: All of your answers should make sense visually!
a) What point on the unit circle is intersected by an angle of
p
4
?
b) What point on the unit circle is intersected by an angle of 210°?
c) What point on the unit circle is intersected by an angle of
5p
?
6
d) What point on the unit circle is intersected by an angle of 315°?
e) What point on the unit circle is intersected by an angle of
4p
?
3
1
f) Find the measure of the angle in both degrees and radians that intersects the point  , 

2
3
 on the unit circle?
2 
g) Find the measure of the angle in both degrees and radians that intersects the point (0,-1) on the unit circle
32



h) Find the measure of the angle in both degrees and radians that intersects the point  
2 2
,
 on the unit
2 2 
circle.
Exercise #4: Use the unit circle and/or reference angles to find the following trigonometric values.
a)
sin 90
c)
e)
cos
5p
3
b)
cos 270
d)
sin
f)
cos p
7p
6
The reciprocal and quotient relationships can be used with the unit circle and/or reference angles to find trigonometric
values. These problems are challenging because they combine much of what has been learned this unit.
g)
h)
i)
j)
tan330°
l)
cot 225°
k)
sec300°
33
The Unit Circle Part 2
Common Core Algebra 2 Homework
1. Explain why the identity sin2 q + cos2 q = 1 is true. Use a picture of the unit circle to support your answer.
2. Use the Pythagorean Identity to verify for acute angle B, if sinB =
4
33
, cosB =
.
7
7
3. Answer the following questions involving points or angles on the unit circle.
a) What point on the unit circle is intersected by an angle of 45°?
b) What point on the unit circle is intersected by an angle of 2p ?
c) What point on the unit circle is intersected by an angle of
7p
?
4
d) What point on the unit circle is intersected by an angle of 240°?
æ 2 2ö
,
÷ on the unit circle.
2
2
è
ø
e) Find the measure of the angle in both degrees and radians that intersects the point ç
f) Find the measure of the angle in both degrees and radians that intersects the point (0,-1) on the unit circle.
34
æ 1
è 2
g) Find the measure of the angle in both degrees and radians that intersects the point ç - ,-
3ö
on the unit
2 ÷ø
circle.
h)
æ1
Find the measure of the angle in both degrees and radians that intersects the point ç ,-
è2
4. Use the unit circle and/or reference angles to find the following trigonometric values.
a)
b)
c)
e)
cos
5p
4
d)
sin
5p
3
f)
sin
5p
6
g)
h)
i)
j)
cot120°
l)
sec225°
k)
csc300°
3ö
on the unit circle.
2 ÷ø
35
Lesson #80 – The Graphs of Sine and Cosine
Common Core Algebra 2
Exercise #1: Bert the beetle is taking a walk counterclockwise
around the unit circle starting from the point (1,0). We are interested
in tracking how far he has travelled and how far above the x-axis he
is.
a) How can his distance travelled and his distance above the x-axis
be defined in terms of what we have learned this unit?
Distance Travelled:
Distance above the x-axis:
b) Fill in the table below to track these two variables. We will split
the total distance travelled into 1/8ths of the distance around the
circle. For each 1/8 of a circle, record her distance to the right of the x-axis, rounded to the nearest tenth when
necessary.
Distance
Travelled
Distance
Above
the xaxis
Graph Herbert’s trip on the axes below using his distance travelled as the independent variable (x) and the height above
the x-axis as the dependent variable (y).
If Herbert continued walking around the unit circle, the pattern would repeat. Continue this pattern on the graph above.
From his starting point, if Herbert walked clockwise around the unit circle, his distance would be negative, and the pattern
would continue in the reverse direction, graphing the points for each quadrantal angle.
The equation of Herbert’s graph is the function, _______.
36
Exercise #2: Sally the spider is taking a walk counterclockwise
around the unit circle starting from the point (1,0). We are interested
in tracking how far she has travelled and how far to the right of the
y-axis she is.
c) How can her distance travelled and her distance to the right of
the y-axis be defined in terms of what we have learned this unit?
Distance Travelled:
Distance to the right of the y-axis.
d) Fill in the table below to track these two variables. We will split
the total distance travelled into 1/8ths of the distance around the
circle. For each 1/8 of a circle, record her distance to the right of the y-axis, rounded to the nearest tenth when
necessary.
Distance
Travelled
Distance
to right
of x-axis
Graph Sally’s trip on the axes below using her distance travelled as the independent variable (x) and her distance to the
right of the y-axis as the dependent variable (y).
If Sally continued walking around the unit circle, the pattern would repeat. Continue this pattern on the graph above.
From her starting point, if Sally walked clockwise around the unit circle, her distance would be negative, and the pattern
would continue in the reverse direction, graphing the point for each quadrantal angle.
The equation of Sallly’s graph is the function, _______.
37
To summarize, we will graph the sine and cosine functions without thinking about a bug walking around a circle. In
the first two exercises, the interval went by
p
p
.
2
Exercise #3: Using the chart, create a graph for y=sin  on the interval 0    2 . Use the unit circle to find the sine
values for the quadrantal angles. Extend the pattern so that     3
Input
Output

Sin 
4
. Now the interval will be by
0

2

3
2
2
a) What is the y-intercept of y=sinθ?
b) What is the maximum value of y=sinθ?
c) What is the minimum value of y=sinθ?
d) The midline of a trigonometric graph is the horizontal axis that is used as the reference line about which the graph
of a trigonometric function oscillates (goes up and down). What is the midline of y=sinθ ?
e) The distance from the midline to the maximum value of sinθ is known as the amplitude.
What is the amplitude of y=sinθ?
f) The period of a trig. function is the interval length of one complete cycle of the graph.
What is the period of y=sinθ?
Exercise #4: Using the chart, create a graph for y=cos on the interval 0    2 . Use the unit circle to find the
cosine values for the quadrantal angles. Extend the pattern so that     3
Input
Output

Cos 
38
0

2

3
2
2
a) What is the y-intercept of y=cosθ?
b) What is the maximum value of y=cosθ?
c) What is the minimum value of y=cosθ?
d) The midline of a trigonometric graph is the horizontal axis that is used as the reference line about which the graph
of a trigonometric function oscillates (goes up and down). What is the midline of y=cosθ?
e) The distance from the midline to the maximum value of cosθ is known as the amplitude.
What is the amplitude of y=cosθ?
f) The period of a trig. function is the interval length of one complete cycle of the graph.
What is the period of y=cosθ?
39
The diagrams give a summary of how the graphs of sine and cosine, respectively, are generated from the unit
circle.
The Graphs of Sine and
Cosine

Sin 
0

2
Common Core Algebra 2 Homework
1.
Use the unit circle to fill out the table for 0£ q £ 2p . Then sketch the graph of
y=sinθ over the interval -2p £ q £ 2p .
a. Name one interval where the graph of y=sinθ is increasing.
b. Name one interval where the graph of y=sinθ is decreasing.
c. What is the maximum value of y on the graph of y=sinθ?
d. What is the minimum value of y on the graph of y=sinθ?
40
e. What is the period of the sine function?

0

2
Cos 
f.
What is the amplitude of the sine function?
g.
What is the midline of the sine function?
h.
Is y=sinθ an even function or an odd function? Explain your answer.
2.
Use the unit circle to fill out the table for 0£ q £ 2p . Then sketch the graph of
y=cosθ over the interval -2p £ q £ 2p .
v
v
v
a. Name one interval where the graph of y=cosθ is increasing.
b. Name one interval where the graph of y=cosθ is decreasing.
c. What is the maximum value of y on the graph of y=cosθ?
41
d. What is the minimum value of y on the graph of y=cosθ?
e. What is the period of the cosine function?
f.
What is the amplitude of the cosine function?
g. What is the midline of the cosine function?
h. Is y=cosθ an even function or an odd function? Explain your answer.
Lesson #81 – Transformations of Sine and Cosine
Common Core Algebra 2
In the
last lesson you learned 3 important
qualities of sine and cosine functions:
midline, amplitude, and period of
graphs. Below is a review of the
definitions of these three qualities.
the
these
The
the
midline of a trigonometric graph is
horizontal axis that is used as the
reference line about which the graph
trigonometric function oscillates (goes
and down).
of a
up
The amplitude of a trigonometric graph is the distance from the midline to the maximum value of the function.
The period of a trigonometric graph is the interval length of one complete cycle of the graph.
Exercise #1: Below are six different transformations of either sine or cosine. Identify the maximum value, minimum
value, midline, amplitude, and period of each.
a)
Maximum
Minimum:
Midline:
Amplitude:
Period:
b) Maximum
Midline:
Period:
Minimum:
Amplitude:
42
c)
Maximum
Minimum:
Midline:
Amplitude:
d) Maximum
Midline:
Period:
e)
Amplitude:
Period:
Maximum
Minimum:
Midline:
Amplitude:
Period:
Minimum:
f)
Maximum
Minimum:
Midline:
Amplitude:
Period:
Exercise #2: Identify each transformation on f(x) in the table below.
43
Function Notation
Transformation
Groups of Transformations
f(-x)
-f(x)
f(x+a)
f(x-a)
f(x)+a
f(x)-a
af(x) where
a 1
af(x) where
0  a 1
f(ax) where
a 1
f(ax) where
0  a 1
a) Which group of transformations in the table would affect the midline of a trigonometric graph?
b) Which group of transformations in the table would affect the amplitude of a trig. graph?
c) Which group of transformations in the table would affect the period of a trigonometric graph?
New Definition:
A horizontal shift of a trigonometric function is known as a phase shift. We will only be identifying phase
shifts in equations.
Formulas
y = asin(b(x + c))+ d or
y = acos(b(x + c))+ d
Midline = d
Amplitude = |a|
Maximum Value = |a| + d
Minimum Value = -|a| + d
2p
Amplitude, Period,b and Frequency
Amplitude: The height of the a trigonometric function from the midline to its maximum y-value.
Period =
Phase Shift = c
æ æ
p öö
Example: y = -5cos ç 3ç x - ÷ ÷ + 9
2øø
è è
The midline, d, is 9. The parent cosine curve has a midline of y=0, so a shift up 10 would result in a midline of 10.
44
The amplitude, |a|. is 5. The parent cosine curve has an amplitude of 1, so a vertical stretch of 5 would result in an
amplitude of 5.
The maximum value is 14 and the minimum value is 4. If there were no vertical shift, the maximum would be 5 and the
minimum would be -5. The vertical shift moves both of these values up 9.
The period is,
2p
2p
, is
. The parent cosine curve has a period of 2 . Since this graph is horizontally compressed by
b
3
3, the period will become 3 times shorter.
The phase shift is , c,
p
2
to the right.
Note: If there is no parenthesis after the trigonometric function, there is an implied parenthesis around anything multiplied
after the variable.
( )
y = cos2x +3 is the same as y = cos 2x +3.
Exercise #3: Identify the midline, amplitude, maximum and minimum values, period, and phase shift.
a)
y = 5sin(3x) + 2
c)
e)
b)
1 
y  2 cos  x 
2 
f (q ) = -3sin (q + p )
d)
y = cos2p x - 4
æp ö
f (x) = - sin ç x ÷
è6 ø
f)
2
f (q ) = - cos(7(q + 3))
3
45
g)
æ æ
p öö
y = -100cos ç 3ç x - ÷ ÷ + 50
2øø
è è
h)
y = 20cos
Transformations of Sine and Cosine
Common Core Algebra 2 Homework
Identify the maximum value, minimum value, midline, amplitude, and period of each.
a)
c)
Midline:
b) Midline:
Amplitude:
Amplitude:
Period:
Period:
Midline:
Amplitude:
d) Midline:
Amplitude:
p
12
x + 30
46
Period:
e)
Period:
(Solid graph, not dotted graph)
Midline:
f)
Midline:
Amplitude:
Amplitude:
Period:
Period:
1. For each of the following trigonometric functions, identify the midline, amplitude, maximum value, minimum
value, period, and phase shift.
a)
c)
y = 4sin(2x) - 3
æ pö
f (q ) = 10sin ç q - ÷
4ø
è
b)
æ1 ö
y = -5cos ç x ÷
è3 ø
d)
æp ö
y = -30sin ç x ÷ + 25
è 15 ø
47
e)
(
)
f (x) = cos 2 ( x + p ) +1
f)
æp
ö
y = 10cos ç (x + 5)÷
è 10
ø
Lesson #82 – Finding the Equation of Transformations of Sine and Cosine
Common Core Algebra 2
This lesson we will be working from the midline, amplitude, and period to find the equations of trigonometric graphs.
First we will review the basic shapes of y=sin() and y=cos() and their reflections in the x-axis, y= - sin() and
y= - cos().
Exercise #1: Use the unit circle to sketch the graph of f(θ)=sinθ on the first grid and f(θ)=cos on the second grid, both
over the interval 0£ q £ 2p .
On the same grid, reflect each function in the x-axis.

f(θ)
g(θ)
48
The y-intercept of every sine curve is ______________________.
From that point the sine curve first
If the sine curve has been reflected over the
increases/decreases (circle one).
increases/decreases (circle one).
x-axis, the sine curve first
The y-intercept of a positive cosine curve
is
________________________.
From that point the cosine curve has to
increase/decrease (circle one).
Therefore, the y-intercept of a cosine curve that has been reflected over the x-axis is _________________________.
From that point the reflected cosine curve has to increase/decrease (circle one).
Exercise #2: Write the equation of each trigonometric function.
Steps:
1. Determine if the function is a positive or negative sine or cosine curve.
2. Determine the midline. This is the value of d in y = asin(b(x + c))+ d or y = acos(b(x + c))+ d .
3. Determine the amplitude. This is the value of a in
if a is positive or negative using step 1.
4. Determine the period.
y = asin(b(x + c))+ d or y = acos(b(x + c))+ d .
2p
= b in y = asin(b(x + c))+ d or y = acos(b(x + c))+ d .
Per.
We will not be working with phase shifts when writing equations of trig functions, so the value of c is 0.
Graph
Work
Determine
49
a)
b)
c)
d)
e)
(Use estimation for this problem).
50
1.
2.
3.
4.
Checking Trigonometric functions on the calculator
Make sure your calculator is in RADIAN mode.
Type the function in Y1.
Set your X-window to include at least one period of the graph OR to match a given graph.
Set your Y-window to include at least the maximum and minimum values of the graph OR to match a
given graph.
Exercise #3: Use the given qualities to write a trigonometric function for each situation. Draw a sketch from your
calculator to support your answer.
a) A cosine curve with an amplitude of 3, a midline of 2, and a period of
b) A sine curve with an amplitude of 6, a midline of -5, and a period of
p
.
12 .
c) A cosine curve with an amplitude of 90, a midline of 100, and a period of
30 .
Transformations of Sine and Cosine
Common Core Algebra 2 Homework
1. Write the equation of each trigonometric function.
Graph
Work
51
a)
b)
c)
Hint: Continue this graph until you have one period.
d)
Estimate for this problem. Assume the period is exactly
12 months.
52
e)
f)
g)
2. Use the given qualities to write a trigonometric function for each situation. Use your calculator to
draw a sketch to support your answer.
a) A sine curve with an amplitude of 4, a midline of 1, and a period of 6p .
b) A cosine curve with an amplitude of 12, a midline of 24, and a period of
c) A sine curve with an amplitude of 6, a midline of 0, and a period of 6 .
p
2
.
53
Lesson #83 – Sinusoidal Modeling
Common Core Algebra II
The sine and cosine functions can be used to model a variety of real-world phenomena that are periodic, that is, they repeat
in predictable patterns. The key to constructing or interpreting a sinusoidal model is understanding the physical meanings
of the coefficients we’ve explored in the last three lessons.
SINUSOIDAL MODEL COEFFICIENTS
For
the midline or average y-value of the sinusoidal model. The midline can be found by finding the average of
the maximum and minimum values of the function.
the amplitude or distance the sinusoidal model rises and falls above its midline. In a word problem, the
amplitude can be found by subtracting the midline value from the maximum value.
the period of the sinusoidal model – the minimum distance along the x-axis for the cycle to repeat
( )
Exercise #1: The tides in a particular bay can be modeled with an equation of the form d = acos bt + d , where t
represents the number of hours since high-tide and d represents the depth of water in the bay. The maximum depth of water
is 36 feet, the minimum depth is 22 feet and high-tide is hit every 12 hours.
(a) On the axes, sketch a graph of this scenario for two full
periods. Label the points on this curve that represent high
and low tide.
d (ft)
(b) Determine the values of A, B, and D in the model. Verify
your answers and sketch are correct on your calculator.
t (hrs)
(c) Tanker boats cannot be in the bay when the depth of water is less or equal to 25 feet. Using the domain, 0 £ t £12 ,
graphically determine the time interval when the tankers cannot be in the bay. Round times to the nearest tenth of an
hour.
54
Exercise #2: The height of a yo-yo above the ground can be well modeled using the equation h  1.75cos  t   2.25 ,
where h represents the height of the yo-yo in feet above the ground and t represents time in seconds since the yo-yo was
first dropped from its maximum height.
(a) Determine the maximum and minimum heights that
the yo-yo reaches above the ground. Show the
calculations that lead to your answers.
(b) How much time does it take for the yo-yo to return to
the maximum height for the first time?
Exercise #3: A Ferris wheel is constructed such that a person gets on the wheel at its lowest point, five feet above the
ground, and reaches its highest point at 130 feet above the ground. The amount of time it takes to complete one full rotation
is equal to 8 minutes. A person’s vertical position, y, can be modeled as a function of time in minutes since they boarded, t,
( )
by the equation f (t) = Acos Bt + D . Sketch a graph of a person’s vertical position for one cycle and then determine the
values of A, B, and D. Show the work needed to arrive at your answers.
y (ft)
t
(min)
Exercise #4: The possible hours of daylight in a given day is a function of the day of the year. In Poughkeepsie, New York,
the minimum hours of daylight (occurring on the Winter solstice) is equal to 9 hours and the maximum hours of daylight
(occurring on the Summer solstice) is equal to 15 hours. If the hours of daylight can be modeled using a sinusoidal equation,
what is the equation’s ampltitude?
(1) 6
(3) 3
(2) 12
(4) 4
55
Sinusoidal Modeling
Common Core Algebra II Homework
APPLICATIONS
1. A ball is attached to a spring, which is stretched and then let go. The height of the ball is given by the sinusoidal equation
 4 
y  3.5cos 
t   5 , where y is the height above the ground in feet and t is the number of seconds since the ball
 5 
was released.
(a) At what height was the ball released at? Show the
calculation that leads to your answer.
(d) Draw a rough sketch of one complete period of this
curve below. Label maximum and minimum points.
y (ft)
(b) What is the maximum height the ball reaches?
(c) How many seconds does it take the ball to return to
its original position?
t (min)
2. An athlete was having her blood pressure monitored during a workout. Doctors found that her maximum blood pressure,
known as systolic, was 110 and her minimum blood pressure, known as diastolic, was 70. If each heartbeat cycle takes
( )
0.75 seconds, then determine a sinusoidal model, in the form y = Asin Bt + D , for her blood pressure as a function
of time t in seconds. Show the calculations that lead to your answer.
56
3. On a standard summer day in upstate New York, the temperature outside can be modeled using the sinusoidal equation
 
O  t   11cos  t   71 , where t represents the number of hours since the peak temperature for the day.
 12 
(b) For 0  t  24 , graphically determine all points in
time when the outside temperature is equal to 75
degrees. Round your answers to the nearest tenth of
an hour.
(a) Sketch a graph of this function on the axes below
for one day.
90
O (degrees F)
50
24
t (hours)
4. The percentage of the moon’s surface that is visible to a person standing on the Earth varies with the time since the
moon was full. The moon passes through a full cycle in 28 days, from full moon to full moon. The maximum percentage
of the moon’s surface that is visible is 50%. (When there is a new moon, 0% of the moon’s surface is visible). Determine
( )
an equation, in the form P = Acos Bt + D for the percentage of the surface that is visible, P, as a function of the
number of days, t, since the moon was full. Show the work that leads to the values of A, B, and D.
5. Evie is on a swing thinking about trigonometry (no seriously!). She realizes that her height above the ground is a periodic
 
t   5 , where t represents time in seconds. Which of the
2 
function of time that can be modeled using h  3cos 
following is the range of Evie’s heights?
(1) 2  h  8
(3) 3  h  5
(2) 4  h  8
(4) 2  h  5
57
Lesson #84 - Revisiting Regression
Common Core Algebra II
The process for finding a trigonometric regression equation, C: Sinreg, is the similar to finding a linear or exponential
regression. The resulting equation from this type of regression is in the form y = asin(bx+c)+d. Notice the slight difference
from the general form we have been working with, y = asin(b(x+c))+d.
Trigonometric regression should be used when the scatter plot shows data that is periodic in nature.
After choosing C:Sinreg, you will see the following menu:
Do not make any changes to the menu unless your data is in different lists than L1
and L2.
Leave iterations at 3 and leave the period blank.
SinReg
Iterations: 3
XList:L1
YList:L2
Period:
Store RegEQ:
Calculate
?: There is no cosine regression. How can a sine regression be used to model data
that more closely resembles a cosine curve?
Exercise #1: The temperature of a chemical reaction changes during the reaction. The temperature was measured every two
minutes and the data is shown in the table below.
Time
(min)
Temp
 C
0
2
4
6
8
10
12
14
16
18
20
35.7
38.9
41.6
42.3
40.8
38.4
36.1
34.2
35.9
39.1
41
(a) Why does it seem like this data might be periodic?
Create a quick scatter plot using your calculator to
verify.
(b) Use your calculator to do a sine regression in the form
y  a sin  bx  c   d . Round all parameters to the
nearest tenth. Graph along with your data to
informally assess the fit of the curve.
(c) According to this model, what is the range in
temperatures the chemical reaction will include?
(d) According to this model, what is the time it takes for
the reaction to complete one full cycle, to the nearest
tenth of a minute?
58
Exercise #2: The maximum amount of daylight that hits a spot on Earth is a function of the day of the year. Taking
x  0 to be January 1st, daylight, in hours, was measured for 12 different days. The measurement was the number of
possible hours of sun from sunrise to sunset.
Day
0
34
68
98
118
134
171
203
274
321
346
Daylight Hours
9.0
9.9
11.5
13.1
14.0
14.6
15.2
14.8
13.1
11.5
9.5
(a) What is the natural period of this data set?
(b) Use your calculator with the period from (a) to find
an equation of the form y  a sin  bx  c   d that
fits this data, then examine the graph of the equation
on the scatter plot. How good is the fit?
(c) What is the maximum amount of daylight hours predicted by the model? Show your calculation.
Exercise #3: The average daily temperature T (in degrees Fahrenheit) is Fairbanks, Alaska, is given in the table. Time t is
measured in months with t=0 representing January 1.
Time
(months)
Temp
(°F )
0.5
1.5
2.5
3.5
4.5
5.5
6.5
7.5
8.5
9.5
10.5
11.5
10.1
-3.6
11.0
30.7
48.6
59.8
62.5
56.8
45.5
25.1
2.7
-6.5
a) Write a trigionometric model, f(t) the temperature t months after January 1st.
b) Your family wants to visit Fairbanks, Alaska during a month where the average daily temperature is always over
40°F . Using your regression, graphically determine the months your family could visit Fairbanks, Alaska.
y (ft)
t
(min)
59
Revisiting Regression
Common Core Algebra II Homework
1. The soil's temperature beneath the ground varies in a periodic manner. A temperature probe was left 3 feet underground
and recorded the temperature as a function of the number of days since January 1st ( x  0 ). The temperatures for 14
days throughout the year are shown below.
Day
5
36 57 94 127 153 192 226 241 262 289 305 337 356
Temp ( F ) 41 37 36 40
48
64
68
66
61
58
49
44
42
40
(a) Find a best fit sinusoidal function for this data set in
(d) If the root of a particular plant species will only
the form y  a sin  bx  c   d . Round all
thrive when the soil temperature
is above 50 F ,
parameters
to
the
nearest
hundredth.
graphically determine the interval of days over
which the plant will thrive, to the nearest day.
y
(b) Based on your model from (a) what are the highest
and lowest
temperature
reached
in the soil?
(c) According
to the
regression,
what is the average soil
temperature?
2. The rise and fall of the tides at a beach is recorded at regular intervals. Their period is about 24 hours, but not exactly.
The depth of a tidal marsh was measured over 3-hour time interval and the data is shown below.
Hours
(since midnight)
Depth
(ft)
0
3
6
9
12
15
18
21
24
5.5
8.0
10.5
11.7
10.8
8.4
5.8
4.3
4.9
(a) Find a sinusoidal model for this data using your calculator. Place it in y  a sin  bx  c   d form. Round all
coefficients to the nearest thousandth.
(b) According to your model, what is the period of the tides in hours, to the nearest tenth of an hour.
3. As the paddlewheel of a steamboat turns, a point on the paddle blade moves so that its distance, h, from the water’s
surface is a sinusoidal function of time. The height of the point at various times is shown in the table.
Hours
(since midnight)
Depth
(ft)
0
3
6
9
12
15
18
21
24
5.5
8.0
10.5
11.7
10.8
8.4
5.8
4.3
4.9
60
a) Based on the fact that the paddle blade is at its highest value in the table at both 4 and 14 seconds, what is a good
estimate of the period of the function modeling this situation?
b) Find the sine regression for this situation with coefficients rounded to the nearest thousandth when necessary.
c) Find the period of the function based on the regression equation. Is this what you expected from part (a)?
d) Is the minimum function value in the table the same as the minimum function value for the regression? Justify your
answer.