Download trigonometric functions and the unit circle

TRIGONOMETRIC FUNCTIONS AND THE UNIT CIRCLE
1. Angles and Angle Measure – pg. 166-179
Assignment: pg. 175-179 #1-9(odd), 11-13(odd)
2. The Unit Circle – pg. 180-190
Assignment: pg. 186-190 #1-5(odd), 6, 10, 12a, 13
3. Trigonometric Ratios I – pg. 191-205
Assignment: pg. 201-205 #1, 3, 4, 6, 8
4. Trigonometric Ratios II – pg. 191-205
Assignment: pg. 202-205 #2, 5, 7, 9-12(odd), 13, 14
5. Trigonometric Equations – pg. 206-214
Assignment: pg. 211-214 #1(odd), 2, 3-5(odd), 6, 7ac, 8-11, 15, 16
6. Graphing Sine and Cosine Functions - pg. 222-237
Assignment: pg.233-237 #1, 2, 4-10, 14, 15, 19
7. Transformations of Sinusodial Functions I – pg. 238-255
Assignment: pg. 250-255 #1-7
8. Transformations of Sinusodial Functions II – pg. 238-255
Assignment: pg. 250-255 #8-18, 23, 24, 27c
9. The Tangent Function – pg. 256-265
Assignment: pg. pg. 262-265 #1-4, 6, 8, 9, 12
10. Equations and Graphs of Trigonometric Functions I– pg. 266-280
Assignment: pg. 275 #1-5
11. Equations and Graphs of Trigonometric Functions II – pg. 266-280
Assignment: pg. 275-279 #6, 9, 10, 12, 13, 15, 16, 19
12. Chapter Quiz
13. Chapter Review (2 days)- pg. 215-217, pg. 282-285
Assignment: pg. 215-217 #1, 2, 5, 7, 9, 11, 13-15, 17, 19-21, 23
pg. 282-285 #3-7, 8, 10-13, 15, 16, 18, 19, 20, 22, 23
14. Chapter Exam
LESSON 1: ANGLES AND ANGLE MEASURE
Learning Outcomes:

To sketch angles in standard position measured in degrees and radians

To convert angles in degree measure to radian measure and vice versa

To determine the measures of angles that are coterminal with a given angle

To solve problems involving arc length, central angles, and the radius in a circle
Need:
 Compass
 Pipe cleaners
 Protractor
Working with a partner:
1. On a separate piece of paper draw a circle with a diameter of 10cm. Label the radius of each
circle as CA.
2. Cut an elastic band to a length equal to radius CA for the circle.
3. Place the elastic band (the one from step 2 above) around the circumference of your circle,
beginning at point A and ending at point P, as shown by the arrow arc below:
P
C
A
4. On the circle you are working with, draw radius CP.
5. Measure ∠PCA with a protractor. What is the measure of ∠PCA in degrees?
2
6. When you create a sector of a circle that has both radii and arc length of the same measure,
you have represented one radian. What is the degree value of 1 radian?
7. What degree measure is represented by a half circle? (see diagram below)
8. Now refer to your labeled circle you drew. Using the elastic band, count how many radians
create a half circle (equivalent to 180º)
9. What famous number is close to your estimation?
10. Using the information above, what conclusion can you make about the radian measure of
180º?
11. What would be the radian measure of 360º?
8. In conclusion, what degree measure is equal to π radians and 2π radians(see definition
below)?
3
The measure of ∠PCA is considered to be 1 radian.
Approximately 60˚
Radian: the measure of the central angle (∠PCA) of a circle subtended by an arc that is the same
length as the radius of the circle.
P
C
θ
A
∠θ = 1
What degree benchmark can we use for 1 radian?
One radian is the measure of the angle at the centre of a circle subtended by an arc equal in
length to the radius of the circle.
Q
O
r
r
P
When a measurement has no units, it is to be considered in radians.
∠POQ = 1 radian
By convention, angles measured in a counterclockwise direction are said to be positive. Those
measured in a clockwise direction are negative.
One full rotation is 360˚ or 2π radians
One half rotation is 180˚ or π radians
𝜋
One quarter rotation is 90˚ or 2 radians
𝜋
One eight rotation is 45˚ or 4 radians
Angle measures without units are considered to be in radians.
Conversion Chart
Degrees to Radians multiply by

180
Radians to Degrees multiply by
4
180

Ex. Draw each angle in standard position. Change each degree measure to radian measure and
each radian measure to degree measure.
a. 30˚
b. -210˚
c.
2
3
5
d. -1.2
Coterminal Angles
Sketch an angle of 60˚ and 420˚ on the same grid. What do you notice?
The terminal arms coincide, these are coterminal angles.
A principle angle is always between 0 and 360 degrees.
Ex. For each angle in standard position, determine one positive and one negative angle measure
that is coterminal with it.
a. 270˚
b. −
5𝜋
4
6
c. 740˚
Need to find principle angle (between 0 and 360)
General Form:
By adding or subtracting multiples of one full rotation, you can write an infinite number of
angles that are coterminal with any given angle.
To find the general form of an angle of 40˚:
40˚+ 360˚ = 400˚
40˚ - 360˚ = -320˚
40˚ + 2(360˚) = 760˚
40˚ - 2(360˚) = -680˚
How can we write this as a general term:
If we wanted to find a general term for the coterminal angles of
2𝜋
3
, how would we write it?
Any given angle has an infinite number of angles coterminal with it, since each time you make
one full rotation from the terminal arm, you arrive back at the same terminal arm. Angles
coterminal with any angle θ can be described using the expression:
𝜃 ± (360°)𝑛 𝑜𝑟 𝜃 ± 2𝜋𝑛,
Where n is a natural number. This way of expressing an answer is called the general form.
Ex. Write an expression for all possible angles coterminal with each given angle. Identify the
angles that are coterminal that satisfy −720° ≤ 𝜃 ≤ 720°, 𝑜𝑟 − 4𝜋 ≤ 𝜃 ≤ 4𝜋
a. -500˚
7
b. 650˚
c.
9𝜋
4
Arc Length:
a  R
where a= arc length
R = radius
 = angle in radians (always!!)
Ex. A Pendulum 30 cm long swings through an arc of 45 cm. Through what angle does the
pendulum swing. Answer in degrees and in radians.
a = 45 cm
a  R
r = 30 cm
8
Ex. Calculate the arc length of a sector of a circle with diameter 9.2m if the sector angle is 150 .
Coordinate Plane (in degrees):
90 ̊
180 ̊
0 ̊ or 360 ̊
270 ̊
Coordinate Plane in radians (both exact and approximate values)
𝜋
2
(1.57)
 (3.14)
0 rad or 2 (6.28)
3𝜋
2
Assignment: pg. 175-179 #1-9(odd), 11-13(odd)
9
(4.71)
LESSON 2: THE UNIT CIRCLE
Learning Outcomes:

To develop and apply the equation of the unit circle

To generalize the equation of a circle with center (0, 0) and radius r

To use symmetry and patterns to locate the coordinates of points on the unit circle
A unit circle is a circle with radius 1 unit:
y
(0,1)
(-1,0)
(1,0)
0
x
(0,-1)
Creating a unit circle. (you will need a blank sheet of paper and a ruler)
1. Draw a circle
2. Label the angles of 30º, 45º, 60º, 90º and all reference angles in each quadrant.
3. Label the equivalent radian measure for each angle.
4. On the back side of the unit circle, draw and label the 45º-45º-90º and 30º-60º-90º special
triangles.
5. Each angle or radian measure on the unit circle can be expressed as a coordinate (cos θ, sinθ).
All the equivalent exact values of each measure can be found in our special triangles. Be sure to
apply the correct positive/negative signs for each measure. Recall the CAST rule to help with
each quadrant.
10
You can find the equation of the unit circle using the Pythagorean Theorem. Consider a point P
on the unit circle. Let P have coordinates (x, y).
y
OP = 1
PA = |𝑦|
OA = |𝑥|
(𝑂𝑃)² = (𝑂𝐴)2 + (𝑃𝐴)²
1² = 𝑥² + 𝑦²
1 = 𝑥² + 𝑦²
P(x, y)
1
0
A
x
The equation of the unit circle is 𝑥² + 𝑦² = 1
The equation for a circle with centre at (0, 0) and radius r is 𝑥² + 𝑦² = 𝑟 2
Ex. Determine the equation of a circle with centre at the origin and radius 6.
Ex. Determine the missing coordinates(s) for all points on the unit circle (always will be radius
of 1 unit) that satisfy the conditions given. Draw a diagram in each case.
2
a. the x-coordinate is 3
Use 𝑥² + 𝑦² = 1
11
Why are there two
answers?
5
b. (𝑥, 13) , 𝑤ℎ𝑒𝑟𝑒 𝑡ℎ𝑒 𝑝𝑜𝑖𝑛𝑡 𝑖𝑠 𝑖𝑛 𝑞𝑢𝑎𝑑𝑟𝑎𝑛𝑡 𝐼𝐼
Recall reference angles.
With a partner, use a diagram to illustrate how we can find the reference angle in each quadrant.
Reference Angles: the acute angle whose vertex is the origin and whose arms are the terminal
arm of the angle and the x-axis.
12
Recall Special Triangles:
sin 60˚ =
√3
sin 30˚ =
1
2
2
cos 60˚ =
cos 30˚ =
1
2
√3
2
tan 60˚ =
tan 30˚ =
√3
1
1
√3
Special Triangles Chart:
We can summarize the exact values of trig ratios of


 




0  0rad  ,30  rad  , 45   rad , 60  rad  ,90  rad  in the following chart:
6

4
3

2

x in degrees
x in radians
0
sin x
0
cos x
1
0
30

6
1
2
45

4
2
2
2
2
60

3
3
2
1
2
90

2
1
0
3
2
tan x
0
1
undefined
3
3
3
𝜋
Ex. a. On a diagram of the unit circle, show all the integral multiples of in the interval 0 ≤
3
𝜃 < 2𝜋.
𝜋
In every quadrant, the reference angle is 6 .
b. Label the coordinates for each point 𝑃(𝜃) on your diagram.
13
y
√3
2
1
1
2
x
In Quadrant 1
In Quadrant 2 :
In Quadrant 3:
In Quadrant 4:
The Unit Circle:
14
Ex. Find the value of cos 120º

Find the value of the reference angle

Find the coordinating value on the unit circle

Consider the quadrant the value is in, will it be positive or negative (CAST rule)

Final answer
Assignment: pg. 186-190 #1-5(odd), 10, 12a, 13
15
LESSON 3: TRIGONOMETRIC RATIOS
Learning Outcomes:

To relate the trigonometric ratios to the coordinates of points in the unit circle

To determine exact values for trigonometric ratios

To identify the measures of angles that generate specific trigonometric values

To solve problems using trigonometric ratios
Recall:
Suppose θ is any angle in standard position, and P(x, y) is any point on its terminal arm, at a
distance r from the origin.
If given the values (x, y) of the two smaller legs of a right
angled triangle, what formula would you use to find the r
value?
Pythagorean Theorem, 𝑟 = √𝑥 2 + 𝑦 2
You can use a reference triangle to determine the three primary trigonometric ratios in terms of
x, y and r.
Knowing that:
Express the sin, cos and tan ratios in terms of x, y and r:
With a unit circle with radius of 1:
𝑥
= 𝑥 , 𝑡ℎ𝑒 𝑓𝑖𝑟𝑠𝑡 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
1
𝑦
sin 𝜃 = = 𝑦 , 𝑡ℎ𝑒 𝑠𝑒𝑐𝑜𝑛𝑑 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒
1
cos 𝜃 =
16
Therefore, you can describe the coordinates of any point 𝑃(𝜃) as (cos 𝜃, sin 𝜃). This is true for
any point 𝑃(𝜃) at the intersection of the terminal arm of an angle θ and the unit circle.
Reciprocal Trigonometric Ratios:
1
cosecant ratio  csc 
sin 
cotangent ratio  cot  
secant ratio  sec 
1
cos
1
tan 
These are reciprocals of the primary trigonometric ratios.
Ex. The point (15, 8) lies on the terminal arm of  as shown. Calculate the
value of r and the exact values of the primary and reciprocal ratios.
15
1 2√2
Ex. The point B(− 3 ,
3
) lies at the intersection of the unit circle and the terminal arm of an
angle θ in standard position.
a. In what quadrant does point B lie?
b. Determine the values of the six trigonometric ratios for θ. Express your answers in
lowest terms.
17
Exact values for the trigonometric ratios can be determined using special triangles.
With the knowledge of the special triangles, the CAST rule can be used to tell us where each
trigonometric ratio is positive.
CAST Rule:
Sine ratio
All ratios
The reciprocal trigonometric ratios
positive
positive
follow the same framework as their
corresponding primary ratio.
Tangent
Cosine ratio
ratio
positive
positive
The trigonometric ratios of any angle can be written as the same function of a positive acute
angle called the reference angle with the sign of the ratio being determined by the CAST rule.
Ex. Determine the exact value for each.
a. sin 210
rotation angle of 210 , has a reference angle of  210 180  30
from special triangles 30 
sin 210  
b. cos
1
. In QIII the sine ratio is negative so
2
1
2
5
3
18
𝜋
c. 𝑡𝑎𝑛 2
d. sec 60°
e. sin(-300˚)
Assignment: pg. 201-205 #1, 3, 4, 6, 8
19
LESSON 4: TRIGONOMETRIC RATIOS II
Learning Outcome:

To relate the trigonometric ratios to the coordinates of points in the unit circle

To determine exact and approximate values for trigonometric ratios

To identify the measures of angles that generate specific trigonometric values

To solve problems using trigonometric ratios
You can determine approximate values for sine, cosine, and tangent using a calculator. Most
calculators can determine trigonometric values for angles measured in degrees or radians. You
will need to set the mode to the correct angle measure.
Check:
Cos 60˚=0.5 (degree mode)
Cos 60 = -0.952412980 (radian mode) (60 radians)
Your calculator can also compute trigonometric ratios for negative angles. However, you should
use your knowledge of reference angles and the CAST rule to check that your calculator display
is reasonable.
Cos (-200˚) = -0.939692620…
Why is this value negative? In what quadrant does the angle terminate?
You can find the value of trigonometric ratio for cosecant, secant, or cotangent using the correct
reciprocal relationship:
sec 3.3 =
1
= −1.0127
cos 3.3
20
Ex. What is the approximate value for each trigonometric ratio. Round your answers to four
decimal places. Be sure to state the quadrant the angle terminates in and why the value is
positive or negative.
a. sin 1.92
b. tan (-500˚)
c. sec 85.4˚
d. 𝑐𝑜𝑠
7𝜋
5
How can you find the measure of an angle when the value of the trigonometric ratio is given?
If sin θ = 0.5, what is the measure of it’s angle?
1
Note: 𝑠𝑖𝑛−1 ≠ 𝑠𝑖𝑛𝜃
The inverse calculator keys return one answer only, when there are often two angles with the
same trigonometric function value in any full rotation. In general, it is best to use the reference
angle applied to the appropriate quadrants containing the terminal arm of the angle.
21
Ex. Determine the measures of all angles that satisfy each of the following.
a. cos θ = 0.843 in the domain 0° ≤ 𝜃 < 360°. Give approximate answers to the nearest
tenth.
b. sin θ = 0 in the domain 0° ≤ 𝜃 ≤ 180°. Give exact answers.
c. cot θ = -2.777 in the domain −𝜋 ≤ 𝜃 ≤ 𝜋. Give approximate answers.
1
1
𝑡𝑎𝑛𝜃 = −2.777, to find the angle 𝑡𝑎𝑛−1 (−2.777) = −0.35. Tan is negative in QII and IV. Use the
reference angle 0.35(the negative means direction in QIV) and apply to the various quadrants.
For 180 to 0: answer in QII is π – 0.35 = 2.79
For 0 to -180: answer in QIV = -0.35 (Recall direction of rotation)
d. 𝑠𝑒𝑐𝜃 =
2
√3
𝑖𝑛 𝑡ℎ𝑒 𝑑𝑜𝑚𝑎𝑖𝑛 − 2𝜋 ≤ 𝜃 ≤ 2𝜋. Give exact answers.
Assignment: pg. 202-205 #2, 5(odd), 7, 9-12(odd), 13, 14
22
LESSON 5: TRIGONOMETRIC EQUATIONS
Learning Outcomes:

Learn to algebraically solve first-degree and second-degree trigonometric
equations in radians and degrees

Learn to verify that a specific value is a solution to a trigonometric equation

Learn to identify exact and approximate solutions of a trigonometric equation in a
restricted domain

Learn to determine the general solution of a trigonometric equation
Investigate:
1
1. What are the exact measures of θ if 𝑐𝑜𝑠𝜃 = − 2 , 0 ≤ 𝜃 < 2𝜋? How is the equation related to
2 cos 𝜃 + 1 = 0?
𝜋
Answers will be in Quadrant II and III. 𝐶𝑜𝑠 −1 (0.5) = 60° 𝑜𝑟 3 .
2. What is the answer for step 1 if the domain is given as 0° ≤ 𝜃 < 360°?
3.What are the approximate measures for θ if 3 cos 𝜃 + 1 = 0 and the domain is 0 ≤ 𝜃 < 2𝜋?
23
4. Complete the chart below. In the left column, show the steps you would use to solve the
quadratic equation 𝑥 2 − 𝑥 = 0. In the right column, show similar steps that will lead to the
solution of the trigonometric equation cos 𝜃² − cos 𝜃, 0 ≤ 𝜃 < 2𝜋.
Quadratic Equation
Trigonometric Equation
𝑥2 − 𝑥
cos 𝜃² − cos 𝜃
5. Identify similarities and differences between solving a quadratic equation and solving a
trigonometric equation that is quadratic.
6. When solving a trigonometric equation, how do you know whether to give your answers in
degrees or radians?
The notation [0, 𝜋] represents the interval from 0 to π inclusive and is another way of writing
0 ≤ 𝜃 ≤ 𝜋.
( means “not included”
[ means “included”
24
Interval Notation:
(diagram)
(description)
Open Interval: (a, b) is interpreted as a < x < b where the
endpoints are NOT included.
(1, 5)
(While this notation resembles an ordered pair, in this context it
refers to the interval upon which you are working.)
Closed Interval: [a, b] is interpreted as a < x < b where
the endpoints are included.
Half-Open Interval: (a, b] is interpreted as a < x <
b where a is not included, but b is included.
Half-Open Interval: [a, b) is interpreted as a < x < b where
a is included, but b is not included.
[1, 5]
(1, 5]
[1, 5)
Non-ending Interval:
is interpreted as x > a where a
is not included and infinity is always expressed as being
"open" (not included).
Non-ending Interval:
is interpreted as x < b where b
is included and again, infinity is always expressed as being
"open" (not included).
We could even show no limits by using this notation: (-∞, +∞)
Ex. Solve each trigonometric equation in the specified domain.
a. 5 sin 𝜃 + 2 = 1 + 3𝑠𝑖𝑛𝜃 , 0 ≤ 𝜃 < 2𝜋
25
b. 4 sec x + 8 = 0, 0 ≤ 𝜃 < 360˚
1
Simplify: sec 𝑥 = −2, to solve cos 𝑥 = − 2
Ex. Solve for θ:
a. tan²θ – 5tanθ + 4 = 0, 0 ≤ 𝜃 < 2𝜋 (answer to nearest tenth)
b. 𝑐𝑜𝑠 2 𝜃 − cos 𝜃 − 2 = 0, 0 ≤ 𝜃 < 360˚
26
For each one of the equations above, is there another way to solve each equation other than
algebraically?
Recall what a general solution is?
Solutions that satisfy a given equation regardless of the amount of rotations apply in either the
positive or negative direction.
Ex. a. Solve 𝑐𝑜𝑠 2 𝑥 − 1 = 0 𝑓𝑜𝑟 0 ≤ 𝜃 < 360˚
b. Determine the general solution for the function where the domain are real numbers measured
in degrees.
If we use coterminal angles for each (add 360 degrees to each):
0: 360, 720, 1080
180: 540, 900, 1260
We notice a pattern, each solution is 180 degrees from each other.
Therefore the general solution is: 𝑥 = 0° + 180°𝑛
We find solutions every 180˚
Assignment: pg. 211-214 #1(odd), 2, 3-5(odd), 6, 7ac, 8-11, 16
27
LESSON 6: GRAPHING SINE AND COSINE FUNCTIONS
Learning Outcome:




To sketch the graphs of y = sin x and y = cos x
To determine the characteristics of the graphs of y = sin x and y = cos x
To demonstrate an understanding of the effects of vertical and horizontal stretches on the
graphs of sinusoidal functions
To solve a problems by analyzing the graph of a trigonometric function
Periodic Functions:
A periodic function is a function whose graph repeats regularly over some interval of the
domain. The length of this interval is called the period of the function.
Graphing Sinusoidal Functions:
 Put calculator in Degree mode
 Use the window x:[0, 360, 90], y:[-2,2,1]
 Using the y-editor, type in y = sin (x)
 This represents one cycle of a sin function
From the graph determine:
1. The x-intercepts
2. The y-intercept
3. Maximum/Minimum values
4. Domain and range
5. What is the connection between the x-intercepts and the unit circle of a sin function?
Now type y = cos (x) in the y-editor using the same window
Determine:
1. The x-intercepts
2. The y-intercept
28
3. Maximum/Minimum values
4. Domain and range
5. What is the connection between the x-intercepts and the unit circle of a cosine function?
The amplitude of a periodic function is defined as half the distance between the maximum and
minimum values of the function.
max  min
amplitude 
2
Graph y  sin  and y  cos  for 0    2
Sin x
Cos x


Must be in radians mode: 0, 4 ,   2, 2,1
2

29
Graph y  2sin  for 0    2
amplitude
2
1

-1
3
2

2
2
-2
Period
For y  a sin 
As a increases or
decreases, the
period stays the
same and the
amplitude increases
or decreases.
a is a vertical
stretch.
Ex. Graph: y=sin x, y=3sin x, and y=0.5sin x, for 0    2 . State the amplitude of each.
y=sin x: amplitude = ,
y=3 sin x: amplitude =
y=0.5 sin x: amplitude =
Ex. Sketch y=cos x and y=cos 2x, for 0    360
y=cos x : 1 complete cycle in
y= cos 2x: Period = 2 complete cycles in
, 1 in
Ex. Now graph y=cos 3x and y=cos0.5x over the same interval. What did you notice?
y=cos 3x: 3 complete cycles in 360 , 1 cycle in
y=cos 0.5x: 1 complete cycle in 720 (360/0.5)
These are all examples of horizontal stretches.
In general:
To find the period of a function in the form y=a sin bx or y=a cos bx:
period =
2
360
(for degrees) period =
(for radians)
b
b
Ex. State the amplitude and period of each:
30
a. y  2 cos 3 : amplitude = , period =
b. y 
1
1
cos  : amplitude =
2
3
, period =
Now we will consider the graphs of the functions whose equations are;
y  a sin b  x  c    d
y  a cos b  x  c    d
and
What transformations are occurring in the following examples:
a.
b.
c.
d.
y= 2 sin x :
y=sin 2x:
y=-3sin x:
y=sin (-3x):
Notice: vertical stretches affect amplitude, and horizontal stretches affect period.
To find the maximum and minimum values use:
d – a and d + a
Ex. a. List the steps involved in graphing the function y = 3 sin 4x.
b. Determine:

Amplitude

Period

Maximum and minimum value
31

x-intercept, y-intercept
sin x has x-intercepts at 0, 180, 360, divide all by 4
x-intercepts will be at 0, 45, 90 in one cycle
y-intercept is 0

Domain and range using interval notation
𝑑𝑜𝑚𝑎𝑖𝑛:
Range:
One cycle of sin x has the following characteristics:
Max value of 1, min value of -1, amplitude of 1, period of 2π, y-intercept of 0, x-intercepts of 0, π
and 2π, domain of 𝜃𝜖𝑅, and a range of −1 ≤ 𝑦 ≤ 1
How many of the parameters above will change if we compare to the graph of y = cos x
Max/min value:
Amplitude/period:
Domain/range:
y-intercept
x-intercepts
Assignment: pg.233-237 #1, 2, 4-10, 14, 15, 19
LESSON 7: TRANSFORMATIONS OF SINUSODIAL FUNCTIONS
32
Learning Outcomes:
 Learn to graph and transform sinusoidal functions
 Learn to identify the domain, range, phase shift, period, amplitude, and vertical
displacement of sinusoidal functions
 To learn to develop equations of sinusoidal functions, expressed in radian and
degree measure, from graphs and descriptions
 To learn to solve problems graphically that can be modeled using sinusoidal
functions
 Learn to recognize that more than one equation can be used to represent the
graph of a sinusoidal function
Recall the equations:
y  a sin b  x  c    d
y  a cos b  x  c    d
and
Ex. Describe how the graph of the functions below compare with y=sinx.
a. y  sin( x  30) :
b. y=sinx + 2:
c. y  sin  x  60 1 :
d. y  45  sin( x  45) :
The vertical displacement can be determined from a graph using the formula:
Max  min
d
2
33
Summary of the Effect of the Parameters a, b, c and d:
y  a sin b  x  c    d
For:
y  a cos b  x  c    d
amplitude = a 
Period =
max  min
2
360
(for degrees)
b
Period =
2
(for radians)
b
Horizontal phase shift = c
 to right if c > 0
 to left if c < 0
Vertical displacement = d


up if d > 0, down if d < 0
max  min
 d
2
Maximum and minimum values:
d + a, d - a
Ex. Write the equations of the following.
a. A cosine function having a horizontal phase shift of 75 right:
b. A sine function having a horizontal phase shift of
displacement 4 units up.
34
3
radians left, and a vertical
5
Ex. Name the amplitude, period, phase shift and displacement of each. How does each
transformation affect the table of values?
2
1
 
a. y   cos  x    3
3
4
12 


b. y  2sin  3x     4  y  2sin 3  x    4
3

Ex. Consider the graph shown, find the sine equation of the graph
Amplitude =
Period =
𝜋
Window [0,2𝜋, 4 ] [−4,4,1]
Assignment: pg. 250-255# 1-7
35
LESSON 8: TRANSFORMATIONS OF SINUSODIAL FUNCTIONS II
Learning Outcomes:
 Learn to graph and transform sinusoidal functions
 Learn to identify the domain, range, phase shift, period, amplitude, and vertical
displacement of sinusoidal functions
 To learn to develop equations of sinusoidal functions, expressed in radian and
degree measure, from graphs and descriptions
 To learn to solve problems graphically that can be modeled using sinusoidal
functions
 Learn to recognize that more than one equation can be used to represent the
graph of a sinusoidal function
Ex. Consider the graph shown.
Write the equation of a cosine function represented by the graph if a) a > 0 and b) a < 0, then a
sine function for a>0.
Window 𝑥: [−
3𝜋 5𝜋 𝜋
4
,
4
, 4 ] , 𝑦: [−5, 5, 1]
36
Sinusoidal Functions:
A function whose graph resembles the sine or cosine curve is called sinusoidal.
Ex. The minimum depth of water in a harbour can be approximated by the function
d(t) = 12 +5 cos 0.5t where 0  t  24
a. Determine the max and min values: Graph the function
b. Period of the function:
c. Suitable Window to view graph:
x : 0, 24, 2 y : 0, 20,5
d. What is the depth of water, at 2:00 am?
e. A ship requires 8.5 m of water in the harbour to dock safely. It is midnight, when will the
boat have to move to prevent grounding?
f. When can the ship return to the harbour?
Assignment: pg. 250-255 #8-18, 23, 24, 27a
37
LESSON 9: THE TANGENT FUNCTION
Learning Outcomes:
 Learn to sketch the graph of y = tanx
 Learn to determine the amplitude, domain, range and period of y = tanx
 Learn to determine the asymptotes and x-intercepts for the graph of y = tanx
 Learn to solve a problem by analyzing the graph of the tangent function
With a partner:
Graph the function y =tan θ, for −2𝜋 ≤ 𝜃 ≤ 2𝜋. Describe as the period, max/min values, the
range, domain, vertical asymptotes x and y intercepts.
𝜋
Window: 𝑥 [−2𝜋, 2𝜋, 4 ] 𝑦: [−5,5,1]
The value of the tangent of an angle θ is the slope of the line passing through the origin and the
point on the unit circle (cosθ, sinθ). You can think of it as a slope of the terminal arm of angle θ
in standard position.
𝑠𝑖𝑛𝜃
𝑦
𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃, 𝑡𝑎𝑛 = 𝑥
When sinθ = 0, what is tan θ?
The value is 0 (x-intercept), sin 180 = 0, x-intercept at 180 degrees or π
When cosθ = 0, what is tan θ? (restriction, when does cos x = 0?)
A vertical asymptote occurs when cosθ=0, this occurs when cos has x-intercepts at 90 and 270
38
For tangent graphs, the distance between any two consecutive vertical asymptotes represents on
complete period.
   
Ex. Graph y= tan 3x for   , ,   3,3,1 and state the amplitude and period.
 2 2 4
Summary of the Effect of the Parameters a, b, c and d:
For y  a tan b  x  c    d
amplitude – not applicable
a value represents:
 a vertical expansion or
 a vertical compression
180
Period =
(degree measure)
b
Period =

b
(radian measure)
horizontal phase shift = c
 right if c > 0, left if c < 0
vertical displacement = d
 up if d > 0
 down if d < 0
39
Ex. A small plane is flying at a constant altitude of 6000m directly toward an observer. Assume
that the ground is flat in the region close to the observer.
a. Determine the relation between the horizontal distance, in metres, from the observer to
the plane and the angle, in degrees, formed from the vertical to the plane.
d
6000 m
plane
𝑡𝑎𝑛𝜃 =
𝜃
observer
𝑑
6000
𝑑 = 6000 tan 𝜃
b. Sketch the graph of the function.
The graph represents the horizontal distance between the plane and the observer. As the plane
flies toward the observer, that distance decreases. As the plane moves from directly overhead to
the observer’s left, the distance value becomes negative. The domain of the function is
−90° < 𝜃 < −90°
c. Where are the asymptotes located in the graph? What do they represent?
Located at 90˚ and -90˚. They represent when the plane is on the ground to the right or left of
the observer, which is impossible, because the plane is flying in a straight line at a constant
altitude of 6000m.
40
d. Explain what happens when the angle is equal to 0˚?
When the angle is equal to 0˚, the plane is directly over the head of the observer. The horizontal
distance is 0.
Ex. Given the coordinate (1, -2.1), determine tan and the value of  in degrees.
Assignment: pg. pg. 262-265 #1-4, 6, 8, 9, 12
41
LESSON 10: EQUATIONS AND GRAPHS OF TRIGONOMETRIC FUNCTIONS
Learning Outcomes:




Learn to use the graphs of trigonometric functions to solve equations
Learn to analyze a trigonometric function to solve a problem
Learn to determine a trigonometric function that models a problem
Learn to use a model of a trigonometric function for a real-world situation
One of the most useful characteristics of trigonometric functions is their periodicity. With a
partner, name as many situations in the world around us that can be represented in a sinusoidal
graph.
Mathematics and scientists use the periodic nature of trigonometric functions to develop
mathematical models to predict many natural phenomena.
You can identify a trend or pattern, determine an approximate mathematical model to describe
the process, and use it to make predictions (interpolate or extrapolate)
You can use graphs of trigonometric functions to solve trigonometric equations that model
periodic phenomena, such as the swing of a pendulum, the motion of a piston in an engine,
motion of a ferris wheel, variations in blood pressure, the hours of daylight throughout a year,
and vibrations that create sounds.
Ex. Determine the solutions for the trigonometric equation 4𝑠𝑖𝑛2 𝑥 − 3 = 0 for the interval 0° ≤
𝑥 ≤ 360° both graphically and algebraically.
42
Algebraically:
Equations involving Multiple Angles:
Consider the equations cos x  
3
3
and cos 2 x  
. Where x is in degrees.
2
2
a. Use a graph to determine the solutions for each over 0  x  360 :
3
2
over: 0  x  360
intersect at:
3
2
over: 0  x  360
intersect at:
graph: cos x  
graph: cos 2 x  
b. Compare:


the number of solutions:
the values of x:
c. Find the general solutions for cos x  
3
:
2
since the period is 360 :
d. Find the general solutions for cos 2 x  
3
:
2
since the period is 180 :
43
Solving a Multiple Angle Equation Using an Algebraic Approach:
1. Find the domain for the multiple angle
2. Solve for the multiple angle between 0 and 2 using the CAST rule and
reference angle.
3. Add the period of the trigonometric graph of the multiple angle to each of
the answers in step 2 until you cover the domain in step 1.
1
Ex. Consider the equation cos 2 x   .
2
a. Find the exact values of x over 0  x  2 :
1
cos 2 x  
2
**Need to find the solutions to cos x  0.5 over 0  x  4 because you will apply the
horizontal compression to your final answer. Your domain is increased because the graph will
condense itself with a horizontal compression.
b. State the general solution:
since the period of cos 2x is  :
Assignment: pg. 275 #1-5
44
LESSON 11: EQUATIONS AND GRAPHS OF TRIGONOMETRIC FUNCTIONS II
Learning Outcomes:




Learn to use the graphs of trigonometric functions to solve equations
Learn to analyze a trigonometric function to solve a problem
Learn to determine a trigonometric function that models a problem
Learn to use a model of a trigonometric function for a real-world situation
Ex. The electricity coming from power plants into your house is alternating current (AC). This
means that the direction of current flowing in a circuit is constantly switching back and forth.
In some Caribbean countries, the current makes 50 complete cycles each second and the voltage
is modeled by 𝑉 = 170 sin 100𝜋𝑡.
a. Graph the voltage function over two cycles. Explain what the scales on the axes represent.
x-axis represents the time passed, the y-axis represents the number of volts
b. What is the period of the current in these countries?
45
c. How many times does the voltage reach 110V in the first second?
50 cycles in one second, need to see how many times it reaches 110 in one cycle.
Graph y = 110 in the second y plot:
Ex. A Ferris wheel ride can be represented by a sinusoidal function. A Ferris wheel has a radius
of 15m and travels at a rate of six revolutions per minute in a clockwise rotation. Ling and Lucy
board the ride at the bottom chair from a platform one metre above the ground.
a.) Sketch three cycles of a sinusoidal graph to represent the height Ling and Lucy are above the
ground, in metres, as a function of time, in seconds. (up and down is one revolution)

What does the radius tell us about the graph?

What is the maximum and minimum height Lucy and Ling will reach?

The Ferris wheel does six revolutions per minute, how long to do one revolution? What
would this represent on our graph or equation?
46
Max at 31
Midline at
y = 16
Min at 1
10
20
1 cycle in 10 seconds,
min to min represents
1 cycle
b. determine the equation of the graph in the form ℎ(𝑡) = acos[𝑏(𝑡 − 𝑐)] + 𝑑
c. If the Ferris wheel does not stop, determine the height Ling and Lucy are above the ground
after 28 seconds. Give answer to nearest tenth of a metre.
d. How long after the wheel starts rotating do Ling and Lucy first reach 12 metres from the
ground. Give answer to nearest tenth of a second.
Assignment: pg. 275-279 #6, 9, 10, 12, 13, 15, 16, 19
47
Cast Rule:
Reference Angles:
Special Triangles:
Trigonometry coordinates:
48
The Unit Circle:
(0,1)
(-1,0)
(1,0)
(0,-1)
49
50