Chapter 4 Trigonometric Functions
4.1 Magnitude of Rotations and Measures of Arcs
Definitions
 Rotation – a _________________________ under which each point in the plane turns a fixed _______________
around a fixed point called the ___________________________________________
o Ex:




Magnitude – the measure of the _____________________________________
Revolution/full turn – when a rotation is ______________
Half turn – when a rotation is ______________
Quarter turn – when a rotation is ______________
o Ex:
Clockwise VS Counterclockwise Rotations
Radian Measure
**Discovering Radians Activity
Converting Between Degrees and Radians
___________ degrees = ______ revolution = _____________ radians
divide by 12 to obtain
___________ degrees = ______ revolution = _____________ radians
Ex 1: The table below lists some of the equivalent measures that result from the basic relationship between degrees
and radians. Copy and fill in the rest of the table.
Degrees
Radians
0°
0
30°
𝜋
6
Revolutions
𝜋
4
1
8
𝜋
3
90°
𝜋
2
120°
3𝜋
4
3
8
150°
180°
𝜋
360°
1
2
1
Ex 2:
a. Convert 1000° to radians exactly
b. Convert 1000° to radians approximately
Ex 3:
a. Convert 1 radian to degrees
b. Explain in words, or with a picture, what 1 radian represents.
Ex 4: Find the length of an arc of 50° central angle I a circle of radius 6 feet.
Circle Arc Length Formula
If s is the length of the arc of a central angle of 𝜃 ____________________ in a circle of radius r, then ______________
Ex 5: A swing hands from chains that are 8 ft long. How far foes the seat of the swing travel if it moves through an angle
of 1.25 radians?
4.2 Sines, Cosines, and Tangents
The UNIT CIRCLE is the circle with center at the _____________________ and radius ___________.
The cosine of 𝜃 (______________) is the ________________________ of P.
The sine of 𝜃 (______________) is the ________________________ of P.
Sine and Cosine Functions are called ______________________________ as well as _____________________________
Ex 1: Evaluate 𝑐𝑜𝑠𝜋 and 𝑠𝑖𝑛𝜋
𝜋
Ex 2: Find the cosine and sine values of the other multiples of 2
Tangent
For all real numbers 𝜃, provided ________________, tan𝜽 = ________________________
Ex 3: Evaluate
a. 𝑡𝑎𝑛𝜋
b. 𝑡𝑎𝑛⁡(−270°)
Ex 4: Suppose the tips of the arms of a starfish determine the vertices of a regular
pentagon. The point A = (1,0) is at the tip of one arm, so is one vertex of a regular
pentagon ABCDE inscribed in the unit circle, as shown at the right. Find the
coordinates of B to the nearest thousandth.
Telling whether 𝒄𝒐𝒔𝜽, 𝒔𝒊𝒏𝜽 and 𝒕𝒂𝒏𝜽 are POSITIVE or NEGATIVE
Ex 5: As of 2008, the largest Ferris wheel in North America is the Texas Star at Fair Park in Dallas, Texas. It seats hand
from 44 spokes. This Ferris wheel is 212 feet tall. How high is the seat off the ground as you travel around the wheel?
4.3 Basic Trigonometric Identities
An identity is an equation that is ___________ for all values of the variables for which the expressions on each side are
defined.
Pythagorean Identity
3
Ex 1: If 𝑐𝑜𝑠𝜃 = 5, find 𝑠𝑖𝑛𝜃
Activity 1!!
Go to the following Desmos website https://www.desmos.com/calculator/88m1bw6vnn
1. Let a point P in the first quadrant be the image of A under the rotation 𝑅𝜃 . Find the values of 𝑐𝑜𝑠𝜃 and 𝑠𝑖𝑛𝜃
from the coordinates of P
2. Reflect P over the x-axis. Call its image Q. Notice that Q is the image of (1,0) under a rotation of magnitude – 𝜃.
Consequently, 𝑄 = (cos(−𝜃) , sin(−𝜃))
a. What are the values of cos(−𝜃)⁡and sin(−𝜃) for your point Q?
b. How are 𝑐𝑜𝑠𝜃 and cos(−𝜃) related? What about 𝑠𝑖𝑛𝜃 and sin(−𝜃) ?
3. Rotate point P 180° around the circle. Call its image H. Notice that H is the image of (1,0) under a rotation of
magnitude (180°+⁡𝜃). Consequently, 𝐻 = (cos(180° + 𝜃) , sin(180° + 𝜃))
a. What are the values of cos(180° + 𝜃) and sin(180° + 𝜃) for your point H?
b. How are 𝑐𝑜𝑠𝜃 and cos(180° + 𝜃) related? How are 𝑠𝑖𝑛𝜃 and sin(180° + 𝜃)?
Opposites Theorem
Half-Turn Theorem
Supplements Theorem
Ex 2: Given that 𝑠𝑖𝑛10° ≈ 0.1736, find a value of x other than 10° and between 0° and 360° for which 𝑠𝑖𝑛𝜃 = 0.1736
Activity 2!!
Using the same Desmos page complete the following:
1. Draw line 𝑦⁡ = ⁡𝑥. Again pick a value of 𝜃 between 0° and 90° and let 𝑃 = 𝑅𝜃 (1,0). Find 𝑐𝑜𝑠𝜃 and 𝑠𝑖𝑛𝜃 for
your value of 𝜃.
2. Reflect point P over 𝑦 = 𝑥 and call its image K. From your knowledge of reflections, what are the coordinates of
point K?
3. In terms of 𝜃, what is the magnitude of the rotation that maps (1,0) onto K. (Hint: K is as far from A along the
circle as P is from the pont (0,1).) Answer in both degrees and radians.
Complements Theorem
1
Ex 3: Given that 𝑠𝑖𝑛30° = , compute the EXACT value of each function below
2
a. 𝑐𝑜𝑠60°
b. 𝑐𝑜𝑠30°
c. 𝑠𝑖𝑛150°
d. 𝑐𝑜𝑠210°
e. sin⁡(−30°)
4.4 Exact Values of Sines, Cosines, and Tangents
Exact Values of Trig Functions when 𝜽 = 𝟑𝟎°, 𝟒𝟓°, 𝟔𝟎°
Ex 1: Use ∆𝑂𝑃𝐹 at the right to compute the exact values of 𝑐𝑜𝑠45° and
𝑠𝑖𝑛45°. Justify your answer.
Ex 2:
a. Derive the exact values of 𝑐𝑜𝑠30° and 𝑠𝑖𝑛30°
b. Find the exact value of 𝑡𝑎𝑛30°
Ex 3: Find the exact values of 𝑐𝑜𝑠120°, 𝑠𝑖𝑛120° and 𝑡𝑎𝑛120°
ACTIVITY! Use your paper plate to copy down every value of the unit circle.
Ex 4: Without using technology, compute the exact value of each trigonometric function below.
𝜋
a. 𝑠𝑖𝑛 4
b. 𝑐𝑜𝑠
5𝜋
6
c. 𝑡𝑎𝑛𝜋
4.5 The Sine and Cosine Functions
Activity 1!!
Complete the in class graphing activity. Be sure to keep the paper as part of your notes.
Activity 2!!
Complete the following table using what you learned in Activity 1.
Sine function
(degrees)
Sine function
(radians)
Cosine function
(degrees)
Cosine function
(radians)
Domain
Range
Zeros
Maxima
Minima
4.6 The Tangent Function and Periodicity
The Tangent Function
𝑠𝑖𝑛𝜃
From the definition 𝑡𝑎𝑛𝜃 = 𝑐𝑜𝑠𝜃, values for the tangent function can be generated
Activity
1. The table below contains some exact values of 𝑡𝑎𝑛𝜃. It also shows decimal equivalents of those values. Fill in the
missing values.
𝜃
0
30° =
𝜋
6
45° =
𝜋
4
60° =
𝜋
3
90° =
𝑡𝑎𝑛𝜃
(exact)
𝑡𝑎𝑛𝜃
(approx.)
𝜃
𝑡𝑎𝑛𝜃
(exact)
𝑡𝑎𝑛𝜃
(approx.)
210° =
7𝜋
6
225° =
5𝜋
4
240° =
4𝜋
3
270°
3𝜋
=
2
𝜋
2
120°
2𝜋
=
3
300°
5𝜋
=
3
135°
3𝜋
=
4
315°
7𝜋
=
4
150°
5𝜋
=
6
180°
=𝜋
330°
11𝜋
=
6
360°
= 2𝜋
2. Graph the values of the tangent function from step 1.
3. Draw a smooth curve through these points, to show the graph of 𝑦 = 𝑡𝑎𝑛𝜃 for all 𝜃, 0 < 𝜃 < 360°, 0 < 𝜃 < 2𝜋
where 𝑡𝑎𝑛𝜃 is defined.
Ex 1: Consider 𝑓(𝑥) = 𝑡𝑎𝑛𝑥
a. Give the domain and range of the function f.
b. Is f an odd function, an even function, or neither? Justify your answer.
A function f is ______________________ if there is a positive number p such that _____________________ for all x in
the domain of f. The smallest such p if it exists, is called the ______________________ of f.
Ex 2: Use the Periodicity Theorem to find:
a. 𝑐𝑜𝑠2670°
b. 𝑡𝑎𝑛1125°
Ex 3: The graph at the right shows normal human blood
pressures as a function of time. Blood pressure is systolic
when the heart is contracting and diastolic when the heart is
expanding. The changes from systolic to diastolic blood
pressure create the pulse. For this function, determine each.
a. the maximum and minimum values
b. the range
c. the period
4.7 Scale-Change Images of Trigonometric Functions
Sine Waves
A pure tone, travels in a ____________________________. The pitch of the tone is related to the _______________ of
the wave; the _________________ the period the ______________ the pitch. The ____________________ intensity of
the tone is related to the _____________________ of the wave. The _______________________ of a sine wave is
____________ the distance between its _________________________ and ___________________________ values.
𝑥
Ex 1: Consider the function with equation 𝑦 = 6cos⁡(3).
a. Explain how this function is related to its parent function, the cosine function.
b. Identify the amplitude and its period.
c. Graph the function.
Ex 2: The graph at the right shows an image of the graph of 𝑦 = 𝑠𝑖𝑛𝑥 under a
scale change. Find an equation for the image.
The ______________________ of a periodic function is the reciprocal of the ___________________ and represents the
number of cycles the curve completes per unit of the independent variable.
Ex:
Ex 3: A tuning fork vibrates with a frequency of 512 cycles per second. The intensity of the tone is the result of a
𝑵
vibration whose maximum pressure is 22 𝒎𝟐. Find an equation to model the sounds wave produced by the tuning fork.
Ex 4: Without using technology, determine how many solutions each equation below has on the interval 0 < 𝑥 < 2𝜋.
Confirm your answer with a graph.
a. cos(3𝑥) = 0.8
𝑥
b. 5 tan (2) = 3
4.8 Translation Images of Trigonometric Functions
Phase Shifts
AKA a ______________________________________________________
Ex 1: Consider the function ℎ with ℎ(𝑥) = sin⁡(𝑥 + 60°). Identify the phase shift.
𝜋
Ex 2: Maximum inductance in an alternating current occurs when the current flow lags behind the voltage by . In a
2
situation of maximum inductance, find an equation for the current, and sketch the two waves. Assume that the two
waves have the same amplitude and period and that the voltage is modeled by the equation 𝑦 = 𝑐𝑜𝑠𝑥.
Phase Shift Identity
Ex 3: Consider the graph of the function 𝑓, where 𝑓(𝑥) = 𝑐𝑜𝑠𝑥
a. Find an equation for its image 𝑔 under the translation (𝑥, 𝑦) → (𝑥, 𝑦 − 2) and sketch the graph of 𝑦 = 𝑔(𝑥)
b. Find the amplitude and period of the function 𝑔
Ex 4: Find an equation for the translation image of the graph of the cosine function shown below
4.9 The Graph-Standardization Theorem
Given a preimage graph described by a sentence in x and y, following processes yield the same graph:
1.
2.
The Graph Standardization Theorem and Trigonometric Functions
𝑦−𝑘
𝑥−ℎ
𝑦−𝑘
𝑥−ℎ
The graphs of the functions with equations
= sin⁡( ) and
= cos⁡( ), with 𝑎 ≠ 0 and 𝑏 ≠ 0, have…
𝑏
𝑎
𝑏
𝑎
Ex 1:
𝑦−1
𝑥+𝜋
a. Explain how the graph of
= cos⁡( ) is related to the graph of 𝑦 = 𝑐𝑜𝑠𝑥
2
3
b. Identify the amplitude, period, vertical shift, and phase shift of this function
Ex 2: Consider the graph of 𝑦 = 2sin⁡(3𝑥 + 𝜋)
a. Describe this graph as the image of the graph of 𝑦 = 𝑠𝑖𝑛𝑥 under a composite of transformations
b. Without graphing, determine the amplitude, period, vertical shift, and phase shift of the sine wave
𝜋
Ex 3: Consider the graph of 𝑦 = 5 cos (2𝑥 − 2 ) − 7
a. Describe this graph as the image of the graph of 𝑦 = 𝑐𝑜𝑠𝑥 under a composite of transformations
b. Without graphing, determine the amplitude, period, vertical shift, and phase shift of the sine wave
Ex 4: The first Ferris wheel was built in 1893 for the Columbian Exposition. The radius of the wheel was about 131 feet
and its center was about 140 feet above the ground. Find an equation that models the height above ground level of a
car at time x (in minutes) assuming the wheel is continually rotating at 9 minutes per revolution, and that the car is
initially at the maximum height.
4.10 Modeling with Trigonometric Functions
Simple Harmonic Motion – motion that can be described using a sine or cosine function
 Each point on the graph corresponds to a location of the pendulum or weight at a particular time
Ex 1: A pendulum swings back and forth in a vacuum. Its distance from an object is captured using a motion detector
for the object. The setup is pictured below at the left; at the right is a graph of the pendulum’s distance from the motion
detector. Write an equation for the distance from the pendulum to the object as a function of time.
Ex 2: When an oven is set to a particular temperature, the heat level rises and falls, actually fluctuating slightly above
and below that level as time passes. Assume that when a particular oven is set to 425°, the oven temperature t in
degrees Fahrenheit m minutes after the burner first shuts off satisfies 𝑡 = 425 + 6cos⁡(0.9𝑚)
a. What are the maximum and minimum temperatures of the oven at this setting?
b. What is the period of this sine wave? What does the period represent?