Chapter 4a – Trigonometry Fundamentals
Assignments, Objectives, Notes
Suggested Assignments
(30% bonus on quiz/homework grade if completed and turned in by test day.)
4.1a
4.2
4.4a
4.1b
4.1c
4.3a
4.3b
p.290: 7-75 every other odd (Angles of rotation in degrees and radians)
p.299: 1-35 odd, 37-42 (Unit Circle sine, cosine, and tangent)
p.318: 1-13 odd, 15-43 every other odd (Trig Functions and Angles of Rotation)
p.292: 79-95 odd, 99 (Arc Length and Sector Area)
p.292: 101, 103, 105, 107, 108 (Linear and Angular Speed)
p.308: 1-25 every other odd, 27, 31, 33-41 odd
p.309: 43-61 odd, 63-67
4.4b p.319: 45, 49, 53, 57, 59-91 odd (More Trig Functions and Angles of Rotation
with calculator trig and finding values of .)
Bonus Review – p.365 1-85 odd
Objectives: Students will be able to…
1. Identify a periodic function and find its midline, amplitude, and period.
2. Draw and label angles of rotation and their reference angles.
3. Convert angle measures between degrees and radians.
4. Draw and correctly label the Unit Circle in degrees and radians with coordinates.
5. Approximate angle measure in degrees and radians.
6. Solve problems involving linear and angular speed.
7. Find values of trig functions for angles that are coterminal with the Unit Circle
angles.
8. State and use the Fundamental Trig Identities to find missing sides of right triangles.
9. Solve problems involving right triangle models.
10. Find values of trig functions for any angle of rotation using the calculator and for any
angle of rotation coterminal with a Unit Circle angle without the calculator.
Notes:
Angles of rotation: A measure of “spin”.
a. Standard Position
b. Initial Side / Terminal Side
c. Positive Rotation vs. Negative Rotation
d. Coterminal
e. Drawing rotations more than
360º
An Example of Negative Rotation
Find two positive and two negative coterminal angles for each angle of rotation.
-315
-210
-120
Draw each angle of rotation in standard position. Determine the quadrant in which each
terminates.
25
-110
225
270
400
Radian Measure
Radians are used
to express the
amount of spin
relative to the arc
length traced by
the spin. An angle
that spins enough
to trace an arc
equal in length to
the radius is an
angle measuring 1
radian.
To find the radian measure of an angle, divide its arc length by its radius.
𝑠
𝜃 = , (𝜃 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑎𝑛 𝑚𝑒𝑎𝑠𝑢𝑟𝑒, 𝑠 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑟𝑐 𝑙𝑒𝑛𝑔𝑡ℎ, 𝑟 𝑖𝑠 𝑡ℎ𝑒 𝑟𝑎𝑑𝑖𝑢𝑠)
𝑟
To convert from degrees to radians and back…
Remember, there are  radians in a 180.
𝐷𝑒𝑔𝑟𝑒𝑒𝑠 𝑅𝑎𝑑𝑖𝑎𝑛𝑠
=
180
𝜋
-or180
𝜋
𝐷𝑒𝑔𝑟𝑒𝑒𝑠 = 𝑅𝑎𝑑𝑖𝑎𝑛𝑠 ∙
𝑅𝑎𝑑𝑖𝑎𝑛𝑠 = 𝐷𝑒𝑔𝑟𝑒𝑒𝑠 ∙
𝜋
180
1. Convert the angles to radians and draw them in standard position.
30
315
115
-410
-210
220
2. Convert the angles to degree measure and draw them in standard position.
2𝜋
3
𝜋
6
3
𝜋
3
1
𝜋
4
−
6
3𝜋
-.25
4
3. Find one positive and one negative coterminal angle for each angle of rotation.
3𝜋
-35
105
4
200
12
𝜋
5
4. If possible find a complementary and supplementary angle for each angle of
rotation.
𝜋
3
2𝜋
5
3𝜋
5
−
𝜋
6
35
122
202
The Unit Circle! Woohoo!
For the purposes of trigonometry, a ray rotating _______ degrees counterclockwise
forms a circle. On a coordinate axis, the endpoint of the ray is at the point (0,0) or the
___________. A circle is divided into approximately ____radians and exactly _____
degrees. 180° = ____ radians.
The radius of a UNIT CIRCLE is ______.
Find the radian measure of each of the 16 major angles on the unit circle below. Label
each point with its degree measure, radian measure, and
coordinates.
The unit circle must be memorized. However, there is a copy inside
the front cover of your textbook for quick reference when studying.
Do you remember SOHCAHTOA and the six trig functions? Use them to answer these
questions.
Find the exact values of the six trig functions for angle .

9
6

15
12
Given the value of one of the angle’s trig functions, sketch a right triangle that includes
the angle and find the other 5 trig function values.
cos 𝜃 =
5
7
sec 𝜃 = 6
sin 𝜃 =
4
17
There are certain properties of the trig functions that are always true. We call them trig
identities. The three basic or “fundamental” identities are…
The Reciprocal Identities
sec 𝜃 =
1
cos 𝜃
csc 𝜃 =
1
sin 𝜃
cot 𝜃 =
1
tan 𝜃
The Quotient Identities
sin 𝜃
tan 𝜃 = cos 𝜃
cos 𝜃
cot 𝜃 = sin 𝜃
The Pythagorean Identities
𝑠𝑖𝑛2 𝜃 + 𝑐𝑜𝑠 2 𝜃 = 1
1 + 𝑐𝑜𝑡 2 𝜃 = 𝑐𝑠𝑐 2 𝜃
𝑡𝑎𝑛2 𝜃 + 1 = 𝑠𝑒𝑐 2 𝜃
You can use these identities to find values of trig functions with having to draw the
triangle or use SOHCAHTOA. For Example…
If the sin = ¾ then the sec = 4/3 by the reciprocal identity.
The cos can be found with the Pythagorean identity.
sin2 + cos2 = 1 so by substitution…(3/4)2+cos2 = 1. Solving for cos gives
cos = ±
√7
.
4
(We will see later how it can be negative.)
The csc = 1/cos so…
4
csc = ±
√7
4√7
= ±
7
The tan = sin/cos so…
3⁄
tan = ± √7 4
⁄
4
3 4
= ±( ∙ )
4 √7
3
= ±
√7
3√7
= ±
7
And the cot = 1/tan so…
cot = ±
√7
3
Find the values of the six trig functions using only the identities.
sec 𝜃 = 2
3
cos 𝜃 = 5
csc 𝜃 = √2
How does SOHCAHTOA apply to angles of rotation and the Unit Circle? SOHCAHTOA
was for right triangles. Can you make a right triangle out of a 150 angle? No. But…
If you turn an angle of rotation into a right triangle by dropping a perpendicular line from
its terminal side to the x-axis you can use SOHCAHTOA.
Consider an angle in standard position, whose terminal side intersects a circle of
radius . We can think of the radius as the hypotenuse of a right triangle:
The point (𝑥, 𝑦) where the terminal side of the angle intersects the circle tells us the
lengths of the two legs of the triangle. Now, we can define the trigonometric functions in
terms of (𝑥, 𝑦) , and r :
And, we can extend these functions to include non-acute angles.
The key is the reference and angle and the reference triangle. To find the reference
triangle draw a perpendicular line from the terminal side of the angle to the x-axis. That
makes the triangle. The reference angle (the angle you use to figure out opposite and
adjacent) is always the acute angle formed by the terminal side and the x-axis.
Let’s find the sine, cosine and tangent of the following angles of rotation. Note that the
radius of each circle is not necessarily 1. Also, note that sometimes the trig functions
may be negative.
(5,12)
(-4, 2)
(-4,-3)
Find the values of the six trig functions for the angles drawn above.
Think about all the possible angles of rotation. In what quadrants will sine be positive?
What about cosine? Tangent?
Find the values of the six trig functions for an angle of rotation, , given one value and a
constraint (condition).
sin 𝜃 =
4
,
5
𝜃 𝑖𝑠 𝑖𝑛 𝑄𝐼𝐼
cos 𝜃 = −
8
17
𝜃 𝑖𝑠 𝑖𝑛 𝑄𝐼𝐼𝐼
tan 𝜃 = −
3
4
𝜃 𝑖𝑠 𝑖𝑛 𝑄𝐼𝑉
csc 𝜃 =
13
5
𝜃 𝑖𝑠 𝑖𝑛 𝑄𝐼
Given that the point is on the terminal side of an angle of rotation, find the six trig
functions for the angle.
(2, 3)
(-5, 2)
(-2, 2)
(3, -4)
1
2
(− 2 , − 3)
Draw each angle of rotation. Find and mark its reference angle. Draw its reference
triangle and find the sine, cosine and tangent of the angle without a calculator.
𝜃=
7𝜋
4
𝜃 = 405°
𝜃=
2𝜋
3
𝜃=−
5𝜋
3
𝜃 = −150°
𝜃=
11𝜋
3
Tangent as slope – An important concept!
If the following line is the terminal side of an angle of rotation in the specified quadrant
find the sine, cosine and tangent of the angle.
y = 3x in Q1.
Note that the line goes through the origin and the slope is rise over run. Draw the line in
Q1. Pick any point on the line, draw the reference triangle and use SOHCAHTOA. Why
does the tangent turn out to be the same as the slope. Will it always?
Try these…
1
𝑦 = 3 𝑥 𝑖𝑛 𝑄3
4𝑥 + 3𝑦 = 0 𝑖𝑛 𝑄4
3
𝑦 = − 4 𝑖𝑛 𝑄2
If the angle of rotation is in the Unit Circle then the hypotenuse of the right triangle is 1
and the leg measures are the coordinates of the point where the angle intersects the
circle.
Putting SOHCAHTOA together with the coordinates and the hypotenuse of 1 leads to
the conclusion…
On the Unit Circle, the x value of each coordinate is the cosine of its angle of rotation
and the y value is the sine. Dividing y/x gives the tangent.
Find sine cosine and tangent for each angle of rotation in the unit circle.
2/3
-11/6
7/4
Notice that there will be
undefined values of tangent,
cotangent, cosecant, and
secant. Where will they occur?
Use the circle to determine the
domain, and range of sine and
cosine. Be careful, the domain
isn’t x and range isn’t y.
How long does it take for the
pattern of values of sine and
cosine to start repeating?
Complete the chart for the values of sine cosine and tangent from the Unit Circle. Write
exact values.
Angle
0
𝜋⁄
6
𝜋⁄
4
𝜋⁄
3
Sine
Cosine
Tangent
Cosecant
Secant
Cotangent
𝜋⁄
2
2𝜋⁄
3
3𝜋⁄
4
5𝜋⁄
6
𝜋
7𝜋⁄
6
5𝜋⁄
4
4𝜋⁄
3
3𝜋
2
5𝜋⁄
3
7𝜋⁄
4
11𝜋⁄
6
Arc Length – Linear Speed – Angular Speed
https://www.youtube.com/watch?v=qUZVv0nr2rc
Once the train gets going, are all parts of the train moving at the same speed?
Linear speed is the amount of distance something moves divided by the time it takes to
move.
-ButAngular speed is the amount of radians an angle of rotation turns through divided by the
time it takes to turn.
Some Formulas to know…
Arc Length: 𝑠 = 𝑟𝜃
𝑠
Linear Speed: 𝑣 = 𝑡
Angular Speed: 𝜔 =
1
𝜃
𝑡
Sector area: 𝐴 = 2 𝑟 2 𝜃
A discussion of linear and angular speed begins with a review of arc length and sector
area.
Arc length is exactly what it sounds like, the length of an arc. The formula for arc length
can be thought of as a percentage of the circumference. For example, take a arc that
measures ¼ of a circle with a radius of 5 meters. The circumference of the circle (2πr) is
10π meters. Since the arc is ¼ of the circle, its length is ¼ of 10π meters or 5π/2
meters.
What if the arc is made by an angle measuring d˚? Well d˚ is d/360 of the circle. So,
take the circumference and multiply by the fractional amount we want. As a formula it
looks something like this.
𝑑°
𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ =
∙ 2𝜋𝑟
360
If we reduce the 2/360 and do a little rearranging of the terms we get this.
𝐴𝑟𝑐 𝐿𝑒𝑛𝑔𝑡ℎ = 𝑑° ∙
𝜋
∙𝑟
180
𝜋
Remember 𝑑° ∙ 180 = 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 and we are going to use the Greek letter θ to represent
the radian measure of an angle. As a result we now have a formula for finding Arc
Length (s) when given θ, the radian measure of an angle.
𝑠 = 𝜃𝑟
Sector Area is the area of a “pizza slice” taken from a circle. Start with the fact that
sector area is a fractional amount of the total area (πr2), and figure out the formula for
finding the sector area given the radian measure of an angle. (Hint the formula is given
above.)
1. Find the length of the arc and the area of the sector subtended by each of the
following central angles.
𝜋
3
𝜋
5
4
7𝜋
6
3
Linear Speed is really just the old d=rt formula. A few things are different in appearance
though. We use v instead r for the rate, or speed, and the distance, d, that we are
moving is the arc length, s. Time, t, is still time.
𝑠
𝑠 = 𝑣𝑡 𝑜𝑟 𝑣 = 𝑡
Angular speed, ω, is a measure of how fast a wheel is rotating, or how many radians it
is turning in a certain amount of time.
𝜃
𝜔=
𝑡
When working problems involving angular speed you should keep in mind that 1
revolution is 2π radians.
A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure the
time it takes for one revolution to be 70 seconds. What is the linear speed (in feet per
second) of this Ferris wheel? What is the angular speed in radians per second? Which
horses are moving at a faster linear speed, the horses at closer to the outside or the
ones closer to the inside?
A Ford Expedition comes standard with tires that have a diameter of 25 inches. If the
owner decided to upgrade to tires with a diameter of 30 inches without having the
onboard computer updated, how fast will the Expedition actually be traveling when the
speedometer reads 75 mph?