1. CHAPTER 4
CIRCULAR FUNCTIONS

2. Section 4-1 Measures of Angles and Rotation
The circular functions are: sine (sin), cosine (cos), and tangent (tan)
In trigonometry, angles are seen as a ray and it’s image after a rotation
A positive rotation is counterclockwise
The initial ray and all of its rotations form a circle (disk)
The angles created can be measured in three ways: revolutions, degrees, and radians

3. 1 revolution = 360°
Ex1. 45° = ____________ revolutions
Ex2. Give another angle (in degrees) with the same ending point as 135°
1 revolution = 360° = 2π radians
You MUST write units for revolutions and degrees (others are implied to be radians)
Ex3. Give another angle (in radians) with the same ending point as
To change from degrees to radians:
To change from radians to degrees:

4. To change from radians to revolutions:
To change from revolutions to radians:
Ex4. Change revolution clockwise to degrees
Ex5. Change revolution counterclockwise to radians
Ex6. Change 100° to radians

5. Section 4-2 Lengths of Arcs and Areas of Sectors
To find the length of an arc on a circle (degrees):
Ex1. Find the length of an arc with 75° central angle and a radius of 6 ft
The Greek letter θ (theta) is frequently used as a variable referring to an angle
Circular Arc Length Formula: If s is the length of the arc of a central angle of θ RADIANS in a circle of radius r, then s = rθ

6. A disk is the union of a circle and its interior
A sector of a circle is that part of a disk that is on or in the interior of a central angle
Circular Sector Area Formula: If A is the area of the sector formed by a central angle of θ RADIANS in a circle of radius r, then
Ex2. Circle M has a central angle of radians and a radius of 10 feet
A) Find the area of the sector created
B) Find the length of the arc created

7. Ex3. Circle N has a central angle of 80° with a radius of 10 feet
A) Find the area of the sector created
B) Find the length of the arc created

8. Section 4-3 Sines, Cosines, and Tangents
If the circle we have been working with has a radius of 1, then it is called the unit circle
One way to write a rotation of magnitude θ for a circle with its center at the origin is
The location of the point on the unit circle after a rotation is given by: (cos θ, sinθ)
It is the location of the point (1, 0) under a rotation of magnitude θ
Always start at (1, 0)

9. You should be able to do certain rotations of (1, 0) in your head (45°, 90°, 180°, 270°)
Ex1. Find
To find the tangent of a function:
Ex2. Find
See table at the top of page 247 (blue)
Ex3. A circle has a radius of 12 feet and a central angle of Find the coordinates of the point.

10. Section 4-4 Basic Identities Involving Sines, Cosines, and Tangents
Pythagorean Identity: For every θ,
Notice where you put the squared symbol for trigonometric functions
Ex1. If , find sin θ
Opposites Theorem: For all θ, a) cos θ = cos (-θ) b) sin (-θ) = -sin θ c) tan(-θ) = -tan θ

11. Supplements Theorem: For all θ (measured in radians),
Complements Theorem: For all θ (measured in radians):
Half-turn Theorem: For all θ (measured in radians),
Ex2. If sin θ = .48, find sin (π – θ)
Ex3. If cos θ = .6
A) Find cos (π + θ) B) Find
Ex4. True or False?

12. Section 4-5 Exact Values of Sines, Cosines, and Tangents
By now you should have your entire sine, cosine, and tangent chart filled out
You need to memorize as much as possible
Ex1. Find the exact value of
The unit circle is a tool to use to find other angles with the same ending point, but you should be able to do this without referring to your unit circle every time

13. Give the exact values of each of the following
Ex2. Ex3.
Ex4.

14. Section 4-6 The Sine, Cosine, and Tangent Functions
Remember that the sine function is the y- coordinate of the image of (1, 0) under a rotation of magnitude θ
Domain of sine = all real numbers
Range of sine = {y: -1 < y < 1}
Sine is positive when 0 < θ < π and it is negative when π < θ < 2π
The maximum value of sine = 1 (when ) and the minimum value = -1 (when )

15. The sine function is an odd function (180° rotational symmetry around the origin)
The cosine function is an even function (reflectional symmetry over the y-axis)
Cosine has the same domain, range, maximum value, and minimum value as sine
Zeros of the sine function are at multiples of π
Zeros of the cosine function are at odd multiples of
Range of the tangent function = all real numbers

16. Domain of tangent = set of real numbers EXCEPT odd multiples of
Zeros of tangent function: multiples of π
Tangent is an odd function
A function f is periodic if and only if there is a smallest positive real number p such that f(p + x) = f(x) for all x. Then p is the period of the function.
A periodic function is one that repeats at regular intervals (that interval is called the period)

17. The circular functions are periodic functions
The period for sine and cosine = 2π
The period for tangent = π
Periodicity Theorem: for all θ, and for every integer n,
Ex1. sin 2.1 ≈ .8632, find two other values of θ with the same sine value
Ex2. tan 50° ≈ 1.192, find two other values with the same tangent value

18. Section 4-7 Scale-Change Images of Circular Functions
The graphs of sine and cosine are both known as sine waves
The longer the period of a function (the more stretched out horizontally it is), the lower the pitch of the sound created
The amplitude is half the distance between the maximum and minimum value of a sine wave (see pg. 272)
If you double the frequency of a sine wave, you raise the pitch by one octave

19. If y = cos x is the parent function, and the image is , then
You can rewrite the equation as
There it is a vertical stretch of magnitude 5
There is a horizontal stretch of magnitude 2
The maximum is 5 and the minimum is -5
The amplitude is 5
The period of the function is 4π (2 · 2π)
The image would be 5 times louder with a lower pitch

20. The functions and have amplitude = │b│ and period = 2π│a│
Open your book to page 274
The frequency of a periodic function is the reciprocal of the period
It is the number of cycles the curve completes per unit of the independent variable
y = cos x has a frequency of
Read example 3 on page 275

21. Ex1.
A) Find the amplitude
B) Find the period
C) Find the frequency
Ex2.
A) Find the equation for the image
B) Find the amplitude
C) Find the period

22. Section 4-8 Translation Images of Circular Functions
Phase shift: the least positive or greatest negative magnitude of a horizontal translation that maps the graph of or onto the
given wave (parent function)
Open your book to page 278 (example 1)
Notice on page 279 in example 2 that a vertical translation keeps the amplitude and period the same
Read bottom of pg. 280

23. Ex1. Parent function is y = cos x
Ex1. Parent function is y = cos x. Find the equation for the image under
You will need to be able to look at a graph and determine the equation
Ex2. A) Graph 2 cycles of the function
B) What is the phase shift from y = cos x?
C) What is the phase shift from y = sin x?

24. Section 4-9 The Graph Standardization Theorem
Rubber band transformation: the composite of scale changes and translations
Open your book to page 286 (orange box)
We will apply this theorem to circular functions
Ex1. Compare y = cos x to
Read example 3 on page 289

25. The graphs of the function with and with a ≠ 0, and b ≠ 0
have amplitude = │b│, period = 2π│a│, phase shift = h, and vertical shift = k
Find each of the following for Ex2 and 3
A) Find the phase shift B) Find the period
C) Find the amplitude
Ex2. y = 2sin(3x + π)
Ex3.

26. Ex4. If the parent function is y = sin x, phase shift = , period = π, and amplitude = 2. Write the equation for the image.

27. Section 4-10 Modeling with Circular Functions
You must use the rules that govern circular function and apply them to real-world situations
Read over all of the examples in this section CAREFULLY before starting the homework
Every example is different and every question is different

28. Ex1. In Houston, the longest daylight period (June 21) is about 845 minutes long, and the shortest daylight period (December 21) is about 613 minutes long. The vernal equinox (March 21) has about 729 minutes of daylight. Assuming a sine wave model for the length of daylight, find an equation which gives the number of minutes of daylight in this city as a function of the day of the year.