Download Dugopoloski`s 3rd Edition Trigonometry §2.1 The Unit Circle

Dugopoloski’s 3rd Edition Trigonometry
§2.1 The Unit Circle & Graphing thru §2.2 The Gen’l Sine Wave
Outline
•
•
•
•
•
•
•
Background Trig Function Information
Unit circle
Relationship between unit circle and background information
6 Trigonometric Functions
Values of 6 Trig Functions
Translations of functions
Graphing Sine & Cosine Functions using translations
Here is some review information:
Initial Side – The ray that begins the rotation to create an angle
Terminal Side – The ray that represents where the rotation of the initial side stopped
Angle – Two rays with a common endpoint
∠ABC
Positive – An angle created by the initial side rotating counterclockwise
Negative – An angle created by the initial side rotating clockwise
What is a Radian?
If an ∠α is drawn in standard position with a radius, r, of 1, then the arc, t, subtended by the
rotation of the ray will measure 1 radian (radian measure was the t measure that we used in Chapter 5 btw)
α
r
r = t = 1 radian
This means that for any angle, α, α = t/r
Note: A radian is approximately equal to 57.296º and 1º ≈ 0.01745 radians. Don’t use approximations to
do conversions!
Why Radians?
∠ 's are not real numbers and radians are, so with radian measure the trig function has a domain
with a real number.
Converting
Because the distance around an entire circle, C= 2πr and C is the arc length of the circle this
means the ∠ corresponding to the ∠ swept out by a circle is equivalent to 2π times the radius:
360º = 2π(radius) thus
π radians = 180º
So, 1° = π/180
Multiply the degree measure by π/180
or
1 radian = 180/π Multiply the radian measure by 180/π; or just substitute 180º for π
Example:
a)
108°
Example:
11π
a)
/12
Convert to Radians
b)
325.7°
c)
-135°
d)
540°
Convert to Degrees
b)
-7π/6
c)
-2.92
Using your calculator to check:
Deg ► Rad
1. Set Mode to Radians [2nd][MODE]
2. Enter # [2nd][APPS] [1]
3. Gives decimal approximation Enter (#)
Rad ► Deg
1. Set Mode to Degree [2nd][MODE]
2. [2nd][APPS] [3] [2nd] [APPS] [4]
The Unit Circle and Points on Circle
The function x² + y² = 1, is the algebraic function that describes a circle with radius = 1.
I will call, t , the arc transcribed by rotating a ray located in the initial position (defined by some
texts as the x-axis between QI and QIV) to a terminal position (rotation is always assumed to be
counter clock-wise). The ordered pair (x, y) describes the position on the unit circle after said
rotation, it is called the terminal point.
The arc t and the angle α are identical except in definition and position when the radius of the
circle is 1 (since α = t/1). The arc lies along the unit circle and the angle is the angle made between
the x-axis (between QI & QIV) and the ray. The angle, α, will be described using radian or degree
measure in this supplemental text. A radian is a portion of the total 2π arc that can be transcribed
in a complete revolution of the initial side.
t = 0, π/2, π, 3π/2 & 2π and their multiples t + n • 2π
(0, 1)
x=0
y=1
r=1
(-1, 0)
(1, 0)
x = -1
y=0
r=1
x=1
y=0
r=1
x=0
y = -1
r=1
(0, -1)
t = 0º, 90º, 180º, 270º & 360º and their multiples t + n • 360º
Terminal Points
The position along the unit circle, resulting from the rotation t, can be described by an
ordered pair – the terminal point. This ordered pair is dependent upon the equation
defining the unit circle:
x2 + y2 = 1
The above picture shows the terminal points for t = 0, π/2, π, 3π/2 & 2π or 0, 90, 180, 270,
360º.
Example:
For a rotation t = π/2 or 90º the terminal point is (0, 1)
For a rotation t = 3π/2 or 270º the terminal point is (0, -1)
Example:
a)
For a rotation t = π or 180º, what is the terminal point?
b)
For a rotation t = 2π or 360º, what is the terminal point?
It can be shown, using the fact that the unit circle is symmetric about the line y = x, that
the terminal point for t = π/4 is (√2/2, √2/2).
Statements
Reasons
1.
The unit circle is symmetric to y = x
1.
Given
2.
t = π/4 is equidistant from (1, 0) & (0, 1)
2.
Midpoint of 0 & π/2
on the unit circle
3.
Terminal point P for t = π/4 lies at the
3.
Symmetry
intersection of y = x and x2 + y2 = 1
4.
x2 + x2 = 1
4.
Follows from 3
& substitution
2
1
5.
x = /2
5.
Multiplication Prop.
6.
x = ±1/√2
6.
Square Root Prop.
1
7.
x = /√2
7.
Quadrant I o.p.’s
8.
y = 1/√2
8.
Since y = x, Given
√2
9.
x = y = /2
9.
Rationalization
10.
Terminal point for t = π/4 lies at
10.
Follows from 1-9
P(√2/2, √2/2) in the 1st quadrant
It can also be shown, through a little more complicated argument that t = π/6, has the
terminal point (√3/2, 1/2).
1.
2.
Statements
Points R(0,1), P(x, y), Q(x, -y)
lie on the unit circle
The arc determined by the terminal
point P(x, y) is t = π/6
3.
PR = PQ
4.
d(P, Q) = √(x – x)2 + (y – -y)2 = 2y
2
2
1.
Reasons
Given
2.
Given
3.
Both subtend arcs of length
π
/3 because Q subtends arc of
π
/2 and P subtends π/6
Distance formula & algebra
4.
5.
d(R, S) =√ (x – 0) + (y – 1) = √2 – 2y
5.
Distance formula, x2 + y2 =1
& algebra
6.
2y = √y – 2y
6.
Follows from 3-5
7.
4y2 = 2 – 2y 4y2 + 2y – 2 = 0 
7.
Algebra
2(2y – 1)(y + 1) = 0  y = -1 or y = 1/2
8.
P is in QI ∴ y = 1/2
9.
Given & Quadrant op’s
9.
x2 + (1/2)2 = 1  x2 = 3/4  x = √3/2
10.
Since P(x, y) is on unit circle,
substitution & algebra & QI
10. P(x, y) = (√3/2, 1/2)
11.
Follows from 1-9 And finally, that that t = π/3, which is nothing more than the reflection of t = π/6 across the
line of symmetry y = x, has the terminal point (1/2, √3/2).
With all of this said, the bottom line that you need to know is what I was taught when I
first took trigonometry. My teacher made us memorize two special triangles and this
helped me throughout my studies. MEMORIZE these triangles. I will quiz you on them
daily for a while.
45/45/90 Right ∆
45°
30/60/90 Right ∆
√2
1
2
60°
1
30°
√3
45°
1
Note: Saying “ONE TO ONE TO THE SQUARE ROOT OF TWO” and “ONE TO TWO TO THE SQUARE
ROOT OF THREE” helped me to memorize them. Recall that the sides of a triangle are in proportion to
the angles, so the larger the angle the larger the side length must be (that helped me to place the correct
side lengths).
You need to memorize the following table for the unit circle
t; α
Terminal Pt.; P(x, y)
Draw a picture of the unit circle with the arc length
0; 0
(1, 0)
& terminal points to the left labeled in QI
π
/6 ; 30º
(√3/2, 1/2)
π
/4 ; 45º
(√2/2, √2/2)
π
/3 ; 60º
(1/2, √3/2)
π
/2 ; 90º
(0, 1)
Converting between Radian and Degree Measure is extremely important and having a copy of the
unit circle broken into equivalent measure is nice.
π/
2;
2π/
3;
3 π/
4;
5π/
6;
(-1/2, √3/2)
(0, 1)
(-√2/2, √2/2)
π/
1
√3
3; ( /2, /2)
π/
√2
√2
4; ( /2, /2)
(-√3/2, 1/2)
π/
√3
1
6; ( /2, /2)
0; (1, 0)
π; (-1, 0)
7π/
6;
11π/
6;
(-√3/2, -1/2)
5 π/
4;
√2
7π/
4;
√2
(- /2, - /2)
1
√3
4 π/
3; (- /2, - /2)
5π/
3π/
2;
(0, -1)
3;
(√3/2, -1/2)
(√2/2, -√2/2)
(1/2, -√3/2)
Note 1: Make all 45° marks by 4ths and make all 30º/60º marks by 6ths and count your way around.
Note 2: Think in terms of x-axis and π/6=30°, π/4=45°, and π/3=60° adding and subtracting your way around
the circle 2π.
Instead of memorizing the entire unit circle, we can instead use the QI information and
the idea of a reference number/angle. To do this we will introduce a concept known as a
reference number (angle). Your book refers to it as t (I’ll call that t-bar sometimes for easy
typing). A reference number is the shortest distance between the x-axis and the terminal
point.
t=
3π
/4
π/
√2
√2
4; ( /2, /2)
t =
π
/4
Now that we have a visual of what a reference number is, we need to be able to find one
without the visual. Here is the process:
Reference Νumber – An arc length, t-bar, is a positive arc length less than π/2 made by
the terminal side and the x-axis.
t
t = t-bar
QI
t-bar
QII t-bar = π – t
t = t-bar
t
t-bar
QIII
t-bar = t – π
t
t-bar
QIV t-bar = 2π – t
For t > 2π or for t < 0, divide the numerator by the denominator and use the remainder
over the denominator as t. You may then have to apply the above methodologies of
finding t-bar.
Example:
a)
c)
Find the reference number for the following (#34 p. 407 Stewart)
t = 5π/6
b)
t = 7π/6
t = 11π/3
t = -7π/4
d)
Lastly, we need to find the terminal point on the unit circle for a reference number, t-bar.
This is done quite simply by using the reference number, quadrant information and
having memorized the terminal points for the first quadrant as shown in the table above
(or in the book on p. 403). The following is the process:
Finding the Terminal Point for a Reference Number
1. Determine the quadrant for which t lies
a) Know that QI (+, +), QII (–, +), QIII (–, –) and QIV (+, –)
2. Use the Reference Number t-bar to determine the terminal point’s coordinates (see table
above or on p. 403 of book)
3. Give appropriate signs to the terminal point’s coordinates according to the quadrant –
see step #1
Example:
Give the terminal point for each part in the last example.
Next we will define the trigonometric functions, review some basic geometry and make
the connection to the special triangles that I asked you to memorize in the last section.
Standard Position of α
P (x, y)
“r” is hypotenuse
“y” is
opposite
α
“x” is adjacent
r = √ x2 + y 2 (Pythagorean Theorem)
r > 0 since it is a distance
(an undirected vector meaning it has no direction)
Based on the α in Standard Position the 6 trigonometric functions can be defined. The
names of the 6 functions are sine, cosine, tangent, cotangent, secant and cosecant.
Because there are many relationships that exist between the 6 trig f(n) you should get in a
habit of thinking about them in a specific order. I’ve gotten used to the following order
and I’ll show you some of the important links.
sin α = opp = y Note: When r = 1, sin α = y
hyp
r
cos α = adj = x
hyp
r
tan α = opp
adj
Note: When r = 1, cos α = x
= sin t = y
cos t
x
cot α = adj =
1
opp
tan t
x≠0
= cos t = x
sin t
y
sec α = hyp
adj
=
1
cos t
= r
x
x≠0
csc α = hyp
opp
=
1
sin t
= r
y
y≠0
y≠0
Note 1: These are the exact values for the 6 trig f(n). A calculator will yield only the approximate values
of the functions.
Note 2: This is both the functions and their reciprocal identities and relations that tie to the coordinate
system (an ∠ in standard position). The definitions given in terms of opposite, adjacent and hypotenuse will help
later.
At this point every text gives the following table to fill in and “memorize” for ease of
finding the values of the trig functions. However, with the MEMORIZATION of the
above triangles and the definitions of the trig functions you won’t have to “memorize”
the table, it will write itself.
Example:
Fill in the following table using the definitions of the trig functions
and the above triangles. Note that your book uses a line for the
undefined values, but I want you to write “undefined”.
Values of the 6 Trig F(n) for t
sin t
cos t
tan t
cot t
sec t
csc t
0
π
/6
π
/4
π
/3
π
/2
Note: Another trick for the sin t and the cos t for these special angles is to do
for the sin t and reverse the order 4, 3, 2, 1, 0 for cos t.
√
/ 2 and fill in 0, 1, 2, 3 & 4
At this point you should also know the domains of the six trig functions. The importance
of this should be obvious once you have completed the above table and see which
functions are undefined at what points. (That is showing you what is not in their domain.)
Domains of 6 Trig Functions
Sine and Cosine {t | t ∈ Real Number}
Tangent and Secant {t | t ≠ (2n+1)π/2, n ∈ I} (odd multiples of π/2 )
Cotangent and Cosecant {t | t ≠ nπ, n∈I} (even multiples of π/2)
In order to find the values of the 6 trig functions for values of t that are not between 0 and
π
/2, the following information is helpful:
Signs & Ranges of Function Values
You don’t have to memorize this, but you at least have to be able to develop it, which is
dependent upon knowing quadrant information and standard position.
y
QII
x < 0, y & r > 0
Sin & csc “+”
This Saying Will Help
Remember the Positive F(n)
QI
x, y & r > 0
All F(n) “+”
x
QIII
x&y<0,r>0
QIV
x&r>0,y<0
Tan & cot “+”
Cos & sec “+”
All
All f(n) “+”
Students
sin & csc “+”
Take
tan & cot “+”
Calculus
cos & sec “+”
Let’s go through the QII information using the definitions of the 6 trig f(n) to see how
this works:
In QII, x is negative (x < 0) while y & r are positive (y, r > 0)
So,
sin = y = + = +
csc = r = + = +
r
+
y
+
cos = x
r
= – = –
+
sec =
r = + = –
x
–
tan =
= + = +
–
cot =
x
y
y
x
= – = +
+
You can always develop the table below using sign information, memorize it or use “All
Students Take Calculus” to help know the signs of the trig functions in each of the
quadrants.
θ in Quad
I
II
III
IV
sin θ
+
+
-
cos θ
+
+
tan θ
+
+
-
cot θ
+
+
-
sec θ
+
+
csc θ
+
+
-
Now, we can take all our newfound knowledge and put it together with our earlier
knowledge.
How to Find the Exact Values of the 6 Trig F(n)
Step 1: Draw the t in the coordinate system creating a ∆ w/ the terminal side and the xaxis. In other words, find the reference number for t.
Step 2: Place t, x & y and find r (using r = √x2 + y2) if you don’t already know a basic ∆
Step 3: Use x, y & r (opp, adj & hyp) w/ definitions to write the exact values of 6 trig f(n)
Note: Once you’ve got sin, cos & tan you’ve got the others due to the reciprocal identitites.
Step 4: Simplify (you’ll need to review your radicals)
Example:
Find the sin t and cos t for the points on the unit circle and use
them to give the coordinates of the terminal points.
(#1 p. 416 Stewart)
π/
t= 4
Example:
For t = 5π/6
(essentially #4 p. 416)
a)
b)
What quadrant would t be in?
Find the reference number
c)
d)
e)
Find sin t
Find cos t
Find tan t
f)
Find csc t
Additional Material to Assist in Graphing Trig Functions
One of the things that will help a great deal in learning to graph the trig functions is an
understanding of translation. I’m going to go over the translation of a quadratic function
to assist you in learning how to graph trig functions.
1)
2)
Every function has a general form of the equation and a graph centered at the
origin.
It is from this general form that translations happen. Think of a translation as
moving the shape formed from the general equation around in space. We can
move the shape up, down, left, right, flip it over or stretch/shrink it. It really gets
fun when we do multiple movements!
a)
Stretching/Shrinking ➞ Multiplies the function value (the y-value) by
a constant
b)
Reflection (Flipping it over) ➞ Multiplies the function by a negative
c)
Vertical Translation (Moving it up/down) ➞ Adds a constant to the function
value (the y-value)
d)
Horizontal Translation (Moving it left/right) ➞ Adds a constant value to the
x-value while still outputting the same y-value
Let’s go through this with the quadratic function:
1)
y = x2
First you must know the basics about the general function
^
a)
This function forms a parabola
with its vertex at (0, 0)
^
Vertex: (0, 0)
b)
c)
The general parabola (being an even function) is symmetric
about the y-axis; the line to which the parabola is symmetric is
called the line of symmetry
The vertical line through the vertex is called the line of symmetry
i)
Relating to being an even function
ii)
For every f(-x) there is an equivalent f(x)
Now, let’s look at a typical table of values that we use to graph this function. This will
assist in seeing the translations of this function.
x
y = x2
0
0
1
1
-1
1
2
4
-2
4
3
9
-3
9
Let’s take our first translation to be the reflection. This simply multiplies the y-value by
a negative.
↓ reflection
Visual of Reflection
x
y = x2
y = –x2
0
0
0
(0, 0)
1
1
-1
-1
1
-1
2
4
-4
-2
4
-4
3
9
-9
v
v
-3
9
-9
Next, take a stretching/shrinking translation. This multiplies the y-value by a negative.
Visually it is like “pulling the parabola up by its ends or pushing it down.” When the
constant is > 1 the parabola is stretched and when it is < 1 but > 0 it is shrunk.
↓ strecthing
Visual of Stretching
2
x
y=x
y = 3x2
← stretched
0
0
0
1
1
3
-1
1
3
←original
2
4
12
-2
4
12
3
9
27
Vertex: (0, 0)
-3
9
27
Next, take the vertical translation. This adds to the y-value. Visually it moves the
parabola up and down the y-axis. When a constant is added to the function, the
translation is up and when the constant is subtracted from the translation is down.
↓ vertical
2
Visual of Vertical Up
x
y=x
y = x2 + 2
← vertical (up)
0
0
2
1
1
3
-1
1
3
←original
2
4
6
-2
4
6
Shifted Vertex: (0, 2)
3
9
11
-3
9
11
(0, 0)
Last, the most difficult translation to deal with in terms of ordered pairs, because it
changes the x not the y-coordinate. This is the horizontal translation. Visually it moves
the parabola to the left or right. When a constant is subtracted from the x-value, the
translation is right and when it is added it is to the left (this is the opposite of what you think,
and it is due to the form that the equations take).
↓ horizontal
Visual of Horizontal Right
x
x′
y = (x - 2)2
0
2
0
1
3
1
original→
-1
1
1
2
4
4
horizontal (right)→
-2
0
4
3
5
9
(0, 0)
-3
-1
9
Shifted Vertex: (2, 0)
↑ there’s notice no
change from y = x2
Because this one is the mind bender, I will also include the left shift.
↓ horizontal
Visual of Horizontal Left
x
x′
y = (x + 3)2
0
-3
0
1
-2
1
-1
-4
1
←original
2
-1
4
-2
-5
4
←horizontal (left)
3
0
9
Shifted Vertex: (-3, 0)
(0, 0)
-3
-6
9
↑ there’s notice no
change from y = x2
Now we can return to the task at hand and apply this same knowledge to trig functions.
To do this we will:
1)
Learn the basic shapes of the trig function graphs
2)
Learn the special names that go along with the translations
3)
See each of the translations apply to the trig function graphs
a)
Recognize the translations from an equation (and eventually the opposite; make
equations from translation recognition)
b)
c)
Focus on tabular values for the translations
Take tabular values to graph the function
Trig Graphs
First let’s look at the two most basic trig function graphs. These are the graphs of the sine
and the cosine. We call the visual representations of the sine and cosine functions
sinusoids. We will be graphing the fundamental cycle of the trig functions. The
fundamental cycle for the sine and cosine function is on the interval [0, 2π]. After we
look at tables and graphs for each of the functions we will get into translations and the
special definitions that go along with the translations.
t
0
π
/2
π
3π
/2
2π
y = sin t
0
1
0
-1
0
Note: Scan from p. 420 Stewart
t
0
π
/2
π
3π
/2
2π
y = cos t
1
0
-1
0
1
Note: Scan from p. 420 Stewart
Notice that I have not included all of the intermediate values in my tables. If you know
the general shape of the curve, the basic shape can be drawn based on the minimum and
maximum values and a point midway between. As you can see, the function values for
sine and cosine are the same values that we see on the unit circle as the y-coordinate for
the sine function and as the x-coordinate for the cosine function. The domain of the
functions are the values that t can take on. We graph one period of each function where
0 ≤ t ≤ 2π unless otherwise stated.
Next we will investigate the moving of these visual representations of the functions
around in space, just as we saw happening with the quadratic function.
If y = f(x) is a function and “a” is a nonzero constant such that f(x) = f(x + a) for every “x”
in the domain of f, then f is called a periodic function. The smallest positive constant “a”
is called the period of the function, f.
Period Theorem:
The period, P, of y = sin(Bx) and y = cos(Bx) is given by
P = 2π
B
The amplitude of a sinusoid is the absolute value of half the difference between the
maximum and minimum y-coordinates.
Amplitude Theorem:
The amplitude of y = A sin x or y = A cos x is |A|.
Note: The amplitude is the stretching/shrinking translation
The phase shift of the graph of y = sin (x – C) or y = cos(x – C) is C.
Note: The phase shift is the horizontal translation
General Sinusoid:
The graph of: y = A sin (k[x – B]) + C or y = A cos (k[x – B]) + C
Is a sinusoid with:
Amplitude = | A |
Period = 2π/k, (k > 0)
Phase Shift = B
Vertical Translation = C
Process For Graphing a Sinusoid Based on Translation
1.
Sketch one cycle of y = sin kx or y = cos kx on [0, 2π/k]
a)
Change 0, π/2, π, 3π/2, 2π into the appropriate values based on 2π/k and
4 evenly spaced values.
2.
Change the amplitude of the cycle according to the value of A
a)
Take all general function values of y and multiply them by A
3.
If A < 0, reflect the curve about the x-axis
a)
Take all values of y from 2a) and make them negative
4.
Translate the cycle | B | units to the right if B > 0 and to the left if B < 0
a)
Add B to the x-values in 1a)
5.
Translate the cycle | C | units upward if C > 0 or downward if C < 0
a)
Add C to the y-values in 3a)
Example:
Graph y = 2 sin (3[x + π/3]) + 1
1.
Find the period & compute the 4 evenly spaced points used to graph
y = sin 3x on [0, 2π/3]
a) The x-values are now 0, (2π/3 • 1/4), (2π/3 • 1/2), (2π/3 • 3/4), 2π/3
or [0, π/6, π/3, π/2, 2π/3] Note: The multiplication by 1/4, 1/2 & 3/4 yields 4 points evenly
space over one period.
2.
3.
4.
Find the amplitude & translate the y-values by multiplication
y = 2 sin 3x
a) Sine y-values are usually 0, 1, 0, -1, 0 so now they are 0, 2, 0, -2, 0
Investigate any possible reflection
A is positive so no change occurs here.
a)
Sine y-values remain as in 2a)
Compute the phase shift (horizontal translation) & translate x-values
Note: The phase shift can’t be read until it is written as k[x – B] – this may require a little
algebraic manipulation
5.
y = 2 sin (3[x – (-π/3)])
a)
Add B to x-values in 1a) They were 0, π/6, π/3, π/2, 2π/3, so
now they are (0 – π/3), (π/6 – π/3), (π/2 – π/3), (2π/3 – π/3)
or [-π/3, - π/6, 0, π/6, π/3]
Translate vertically by adding to the last translation of the y-values from
either 2a) or 3a)
y = 2 sin (3[x + π/3]) + 1
a)
The y-values from 2a) or 3a) have 1 added to them.
They were 0, 2, 0, -2, 0 so they are now 1, 3, 1, -1, 1
This is a table that shows the translations. You would plot the points from the furthest
right x & furthest right y.
Plot
Plot
t for sin t
0
π/2
π
3π/2
2π
t′ for sin 3t
0
π/6
π/3
π/2
2π/3
⇓
t′′ for sin(3[t – (-π/3)]
- π/3
-π/6
0
π/6
π/3
Y for sin t
0
1
0
-1
0
Y for 2 sin 3t
0
2
0
-2
0
And here is your graph with the translation from y = sin 3x to f(x)
Note: This example is from Precalculus Functions & Graphs and Precalculus with Limits, Mark
Dugopolski
Example:
a)
b)
What translation is being done to
y = 3 + sin x?
Give a table using the same values of x for y = sin x &
y = 3 + sin x
c)
Sketch both sinusoids on the same graph
(Similar #2 p. 429 Stewart)
⇓
Y for f(t)
1
3
1
-1
1
Example:
a)
b)
c)
d)
e)
Example:
a)
b)
c)
d)
e)
Example:
a)
b)
c)
d)
e)
f)
What translation(s) are being done to y = cos 3x?
What is the amplitude of the function?
What is the period of the function?
Give a table using the same values of x for y = cos x &
y = cos 3x
Sketch both sinusoids on the same graph
What translation(s) are being done to y = 5 sin 1/4x?
What is the amplitude of the function?
What is the period of the function?
Give a table using the same values of x for y = cos x &
y = 5 sin 1/4x
Sketch both sinusoids on the same graph
What translation(s) are being done to y = -4 cos 1/2(x + π/2)?
What is the amplitude of the function?
What is the period of the function?
What is the phase shift?
Give a table using the same values of x for y = cos x &
y = -4 cos 1/2(x + π/2)
Sketch both sinusoids on the same graph
Example:
You can’t see the period or the phase shift as this function is
currently written. Re-write the function and give the period and
the phase shift. (Adapted from #39 p. 429 Stewart)
y = sin (π + 3x)
Using a Graphing Calculator to View a Sinusoid
1)
Find the period, amplitude and phase shift and use these to set the viewing
window (See p. 426 of Stewart’s text discusses the dangers of a viewing window that is not
correctly set. The summary is you need a viewing window that reflects the period and amplitude.)
2)
3)
4)
5)
Put your calculator in RADIAN mode
Use the Y = key to enter the trig function; use X,T, θ, n key for the t value
ZOOM trig to graph
Use WINDOW to set the view rectangle appropriately
Example:
a)
b)
c)
*d)
Let’s try this together and at the same time demonstrate 1)
translations and 2) inappropriate viewing rectangle.
sin 50x
-1 + sin 50x
0.5 sin 50 x
sin 50x – 1
*This is the example that could create havoc if you try to create the wrong viewing window to see all of
these at the same time.
Application of Sinusoids
Frequency is a real world interpretation of the period of a sinusoid. The frequency is the
reciprocal of the period. It tells us how many cycles are in one interval of time. A cycle is
what we see as the defining characteristics of a period – the highs, lows and middle. This
is interpreted in physics for light, sound and energy. You will see applications ranging
from music to electrical engineering and astronomy.
F = 1/P
or
F = k/2π for y = A sin (k[x – B]) + C or y = A cos (k[x – B]) + C
Example:
Give the frequency of the y = cos (0.001πx)
where x is in seconds
* Trigonometry, Dugopolski, 3rd Edition, #42 p.127
Curve Fitting for Sinusoids
Steps to Take
1.
Plot the points given on a coordinate system.
2.
Decide if your graph appears to look more like a sine or cosine function and start
with the appropriate sine function in the form:
y = A sin (k[x – B]) + C or y = A cos (k[x – B]) + C
3.
Find the amplitude A
A = 1/2 (maximum y – minimum y)
4.
5.
6.
7.
Find the vertical shift B
B = 1/2 (maximum y + minimum y)
Find the period k
k = 2π ÷ [ 2(x of maximum y – x of minimum y)]
Find the phase shift C
C = x of the maximum y
Put the A, B, k & C appropriately into the sine function in #1
Example:
Use a sine function to model the following. We are using a sine
because a plot of the points reveals a curve that looks like a sine
wave.* Precalculus: Mathematics for Calculus, Stewart, 5th ed., p. 464
Circadian rhythm is the daily biological pattern by which body
temperature, blood pressure, and other physiological variables
change. The data in the table below show typical changes in
human body temperature over a 24-hour period where t = 0
corresponds to midnight.
Time After Midnight
Body Temperature ºC
0
36.8
2
36.7
4
36.6
6
36.7
8
36.8
10
37.0
14
37.3
16
37.4
18
37.3
20
37.2
22
37.0
24
36.8
Modeling Sinusoids Using Your Calculator
1.
Input data into the STAT Editor into 2 registers
STAT  ENTER  Input Independent Values in L1
Use right arrow key to move to L2 and Input Dependent Variables
2.
Make sure that you are using radian mode
3.
Find the SinReg function. You might find it in STATTESTS or you might need
to use the CATALOG of functions.
2nd 0 then ALPHA LN then use arrow down to find SinReg in CATALOG
4.
Give the registers for the independent and dependent variables
2nd 1 if you used L1 for indep , 2nd 2 if you used L2 for dependent
5.
Read the values for the sine and write appropriately.
6.
If comparing to a hand derived cosine function remember to subtract π/2 (approx.
1.57) from the calculator’s “c” value and then factor so you can see the same
function that you arrived at by hand.
7.
If comparing to a hand derived sine function don’t forget to factor.
*Here is a link to someone else’s calculator handout that I found on the internet. (Here is the link so you
can type it in as well:
http://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=2&ved=0CDkQFjAB&url=http%3
A%2F%2Fwww.austincc.edu%2Fjthom%2FSineRegressionOnTIs.pdf&ei=6vnSUtD7MabQsATKzIGAD
A&usg=AFQjCNFsJS1KCsu2KCzIYldLGyPhBob_g&sig2=Vv9gKOJQoVF0ix1eNCE8NA&bvm=bv.59026428,d.cWc)
Example:
Use your calculator to model the circadian rhythm in the last
example.
Example:
Here is a cosine to model. These are average temperatures for
Montgomery County, Maryland. Use Month = 0 for January for
easier modeling. * Precalculus: Mathematics for Calculus, Stewart, 5th ed.,
p. 463
Month
January
February
March
April
May
June
Ave Temp ºF
40.0
43.1
54.6
64.2
73.8
81.8
Month
July
August
September
October
November
December
Ave Temp ºF
85.8
83.9
76.9
66.8
55.5
44.5