PRECALCULus
Unit 4A
TRIGONOMETRY
PRANGE/JOHNSON
1
Blank page
2
Trigonometry Unit 4A
Learning Target
Practice for the
Learning Target
Section 4-3 is a review of right triangle trig from
Geometry and AAT. It will not be directly assessed
at any time, but the skills are prerequisite to the
following learning targets.
worksheet in this
packet
4.1.1
I can convert between radian and degree measures.
worksheet in this
packet
4.1.2
I can find co-terminal angles.
worksheet in this
packet
4.1.3
I can sketch angles in standard position.
worksheet in this
packet
4.2.1
I can evaluate all 6 trigonometric functions of
radians less than or equal to 2π using the unit circle.
Pg 264 1-27 odd
4.2.2
1 can evaluate all 6 trigonometric functions of
radians greater than 2π using the unit circle.
Pg 264 29-35 odd
4.2.3
I can use a calculator to evaluate trigonometric
functions.
Pg 265 43-51
4.2.4
I can evaluate a trigonometric function that models a
real world situation.
Pg 265 58, 60
4.4.1
I can find the reference angle for any given angle.
4.4.2
Given the value of 1 trig function and domain
constraints, I can find the values of the remaining 5
trig functions.
Pg 284
1,5,7,13,15, 17-23
odd, 67-73 odd
4.4.3
I can evaluate trigonometric functions of any angle.
Angle measures will be in degrees or radians,
positive or negative, acute or obtuse.
Pg 285 29-35 odd,
53-65 odd
4.4.4
I can find 2 solutions to a simple trig equation.
Pg 285 91- 95 odd
section 4-3
Score on
Learning
Target Quiz
Help
needed?
yes/no
Pg 285 37-43 odd
Essential Questions for the chapter:
1. How are the six trigonometric and circular functions related to each other?
2. How do trigonometric and circular functions model real-world problems and their solutions?
Essential Questions for the course
1.
2.
3.
4.
5.
6.
How is this similar or different from what I have done before?
What can I do to retain what I have learned?
Does my answer make sense? If not, what do I do?
Do I need help, and where do I go to find it?
How would a calculator make this problem easier to do?
How do I explain or justify my work to myself and others?
7. What is the given information and how do I use it?
3
LEARNING TARGET QUIZ SCORING RUBRIC
4 MASTERY
I completely understand the strategy and mathematical operations to be used,
and I used them correctly.





My work shows what I did and what I was thinking while I worked the problem.
The way I worked the problem makes sense and is easy for someone else to follow.
I followed through with my strategy from beginning to end.
My explanation and work was clear and organized.
I did all of my calculations correctly.
3
DEVELOPING MASTERY
I completely understand the strategy and mathematical operations to be used,
but a minor error kept me from completing the problem correctly.
2
BASIC UNDERSTANDING
I used mathematical operations and a strategy that I think works for most of the
problem.





1
MINIMAL UNDERSTANDING
I wasn’t sure which mathematical operations to use, and my plan didn’t work.

0
Someone might have to add information for my explanation to be easy to follow.
I know which operations I should have used, but couldn’t complete the problem.
I think I know what the problem is about, but I might have a hard time explaining it.
I’m not sure how much detail I need in order to help someone understand what I did.
I made several calculation errors.
I tried several things, but didn’t get anywhere.
NO EVIDENCE
I left the problem blank.



I didn’t know how to begin.
I don’t know what to write.
I provided no evidence of understanding.
4
4-3 Warm Up(s)
5
Date _______
Notes: 4-3 (REVIEW FROM AAT AND GEOMETRY)
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Previously Learned Skill:
1. Evaluate trigonometric functions using a right triangle.
Vocabulary:
sine
cosine
tangent
cosecant
secant
cotangent
SOH-CAHTOA
opposite leg
adjacent leg
hypotenuse
theta
special right
triangles
30-60-90
45-45-90
7, 24, 25
8, 15, 17
9, 40, 41
Pythagorean
Theorem
2
𝑎 + 𝑏2 = 𝑐 2
Pythagorean
Triple
⇒
3, 4, 5
5, 12, 13
Write the ratios for each of the six trigonometric functions? Use the triangle below to help you.
MEMORIZE
hyp
sin  
csc 
cos 
sec 
tan  
cot  
opp
adj
Example 1: Round to the nearest hundredth, if needed.
a. Solve for x:
b. Solve for r:
r
12
20
45
60
x
6
c. Solve for  :
(You Try) d. Solve for m:
6

30
10
6
m
(You Try) e. Solve for  :
(You Try) f. Solve for  :
14
5.7
48
14

Example 2:
a.

50
Sketch a right triangle that corresponds to the trigonometric function below.
Then find the other 5 trig functions.
cos  
5
7
b. tan   5
7
5
(You Try) c. sin 𝜃 = 13
MEMORIZE
Special Right Triangles – Find the exact values of the remaining sides of the triangles.
30
45
45
60
x
x
Fill in each trig value:
sin 30 
sin 60 
sin 45 
cos 30 
cos 60 
cos 45 
tan 30 
tan 60 
tan 45 
Example 3: Find the value of  in degrees and radians without using a calculator.
a. cos  
c.
2
2
tan   3
b.
sin  
1
2
d. tan   1
8
What is sin θ?
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
Compare and contrast sine and cosine.
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
9
Homework Worksheet for section 4-3
10
Homework Worksheet for section 4-3
11
4-1 Warm Up(s)
12
Date _______
Notes: 4-1
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can convert between radian and degree measures.
2. I can find co-terminal angles.
3. I can sketch angles in standard position.
Vocabulary:
radian
terminal side
Degree
co-terminal
quadrant
standard position initial side
Review of Quadrants & Degree Measures
Measuring Angles: There are two ways to measure angles using degrees & radians. Radians are
often used in calculus.
Definition of a Radian:
s=r

r
13
Things to think about:
1. How many radians are in a circle?
2. How many radians are in half a circle?
14
Degree Measure: There are 360 in one revolution of a circle. Because 2 radians corresponds
to one revolution, degrees and radians are related.
Conversions between Degrees & Radians:
1.
2.
Example 1: Let’s discover how degrees and radians relate …
Step 1, graph the given degree.
𝑑𝑒𝑔𝑟𝑒𝑒
Step 2, make a fraction ( 180∘ ) and reduce.
Step 3, convert from degrees to radians.
Step 4, compare. What patterns are you finding?
a. 135
b. 120°
(You Try) c. 144°
15
_____________________________ are generated by counterclockwise rotation.
_____________________________are generated by a clockwise rotation.
d. -135°
e. -225°
(You Try) e. -150°
Example 2: Convert from radians to degrees.
a. 3𝜋
c. 2 radians
b.
9𝜋
2
𝜋
d. − 2
16
e.
g.
15𝜋
(You Try f – h) f. −4𝜋
6
28𝜋
h. 4 radians
15
MEMORIZE
An angle is determined by rotating a ray about its endpoint. The starting position
of the ray is the ______________________ of the angle and the position after rotation is the
______________________.
An angle is in _______________________ if it fits a coordinate system in which the origin is
the vertex and the initial side coincides with the positive x-axis. Sketch a picture.
Example 3: Sketch each angle in standard position.
a. 135°
b. 400°
c. -120°
d. -600
17
d.
5𝜋
e.
6
h. −
2𝜋
7𝜋
f.
4
i. −
3
7𝜋
g. 6 radians
3
7𝜋
j. −
4
7𝜋
6
Example 4: Determine the quadrant in which the angle lies. The angles in a-d are given in radians.
a.
7𝜋
5
c. -2 radians
(You Try) b. −
11𝜋
9
d. 2.25 radians
18
MEMORIZE
Angles that have the same initial and terminal sides are called _____________________ angles.
Examples of co-terminal angles:
𝝅
𝟔
𝒂𝒏𝒅
𝟏𝟑𝝅
𝟔
60° and 420°
To find co-terminal angles add or subtract 2 or 360° which is one full revolution.
Example 5: Determine two co-terminal angles in radian or degree measure (one positive & one
negative) for each angle.
a. 114°
d. 𝜃 =
b. -390°
5𝜋
4
e. 𝜃 = −
(You Try) c. -280°
13𝜋
4
(You Try) f. 𝜃 = −
11𝜋
6
19
How will you remember what co-terminal means?
___________________________________________________________________________________
___________________________________________________________________________________
___________________________________________________________________________________
Compare and contrast degrees and radians.
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
20
Homework for section 4-1
21
Homework for section 4-1
22
Homework for section 4-1
23
4-2 Warm Up(s)
24
Date _______
Notes: 4-2
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
2. How do trigonometric and circular functions model real-world problems and their solutions?
Learning Targets:
1. I can evaluate all 6 trigonometric functions of radians less than or equal to 2π using the unit
circle.
2. I can evaluate all 6 trigonometric functions of radians greater than 2π using the unit circle.
3. I can use a calculator to evaluate trigonometric functions.
Vocabulary:
unit circle
period
sine
cosecant
cosine
secant
tangent
cotangent
MEMORIZE at least the first quadrant of the UNIT CIRCLE. Save the card stock copy in your
binder.
MEMORIZE
Each point on the coordinate plane is equal to ( ___ , ___ ).
Each point on the unit circle is equal to ( ___________, ___________ ).
Formulas for the 6 Trigonometric Functions:
sin  =
csc  =
cos  =
sec  =
tan  =
cot  =
Example 1:
Evaluate the six trigonometric functions of  given (x, y).
 3 1
, 

2
 2
25
Example 2:
a)
Find the point (x, y) on the unit circle that corresponds to the real number  .
5

b)


4
3
(You Try) c)  
5
3
(You Try) d)   
Example 3: Evaluate the sine, cosine, & tangent of the real number (if possible).
11
a)
b)
t  2

6
(You Try) c)
t
3
4
Example 4: Evaluate (if possible) the six trigonometric functions of the real number.
5
3
a)
b) t 

3
2
26
c)
t
7
4
(You Try) d)  
5
6
MEMORIZE
In general…….the sin(t  2 )  sin t and cos(t  2 )  cos t because adding 2 just adds a
revolution around the circle. If you keep repeating the revolutions you can obtain similar results.
So, sin(t  2 n)  sin t and cos(t  2 n)  cos t for any integer n. Functions that behave in such a
repetitive (or cyclic) manner are called ___________________________________.
Example 5:
Evaluate the trigonometric function using its period as an aid.
9
4
a)
cos 7
b)
sin
c)
 19 
sin  

 6 
d)
 8 
cos  

 3 
Explain why you only need to memorize Quadrant I of the unit circle. How does Quadrant I help you
find any other value on the unit circle?
27
Example 6: Use a calculator to evaluate the expression. Round your answer to three decimal places.
(Your calculator mode must be in radians.)

a)
cos (-2.5)
b)
d)
sec 1.8
(You Try) e) sin (-13.4)
tan
2
c)
cot 3.7
Example 7 - Application Problems:
a) Electric Circuits
The initial current and charge in an electrical circuit are zero. The
current when 100 volts is applied to the circuit is given by
𝐼 = 5𝑒 −2𝑡 𝑠𝑖𝑛 𝑡
where the resistance, inductance, and capacitance are 80 ohms, 20 henrys, and 0.01 farad,
respectively. Approximate the current (in amperes) t = 0.7 second after voltage is applied.
b) Harmonic Motion The displacement from equilibrium of an oscillating weight suspended
by a spring is given by
1
𝑦(𝑡) = 4 cos 6𝑡
where y is the displacement in feet and t is the time in seconds (see figure). Find the
1
1
displacement when (a) t = 0
(b) t = 4
(c) t = 2
28
Homework for 4-2
Pg 264 1-27 odd
Pg 264 29-35 odd
Pg 265 43-51
Pg 265 58, 60
29
Continue homework for 4-2 on this extra page
Pg 264 1-27 odd
Pg 264 29-35 odd
Pg 265 43-51
Pg 265 58, 60
30
4-4 Warm Up(s)
31
Date _______
Notes: 4-4
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can find the reference angle for any given angle.
2. Given the value of 1 trig function and domain constraints, I can find the values of the remaining
5 trig functions.
3. I can evaluate trigonometric functions of any angle. Angle measures will be in degrees or
radians, positive or negative, acute or obtuse.
4. I can find 2 solutions to a simple trig equation.
Vocabulary:
1. Reference angle
Definition of a Reference Angle: Let  be an angle in standard position. Its reference angle is the
acute angle  ' formed by the terminal side  and the horizontal axis.
Quadrant II
𝜃 ′ = _________________________
Quadrant III
𝜃 ′ = _________________________
Quadrant IV
𝜃 ′ = _________________________
Example 1: Find the reference angle.
a.
300°
b.
-135°
c. 240°
32
d.
2𝜋
3
e.
−
7𝜋
f.
6
4𝜋
3
The signs of trigonometric functions in the 4 quadrants
Quad II
Quad III
Quad I
Quad IV
Example 2: The point (3, -4) is on the terminal side of 𝜃. Find sin  , cos  and tan  .
33
Example 3: Determine the exact values of the six trig functions of angle 𝜃.
Example 4: The point (-5, -2) is on the terminal side of an angle in standard position. Determine the
exact values of the six trig functions of the angle.
Example 5: Determine the quadrant in which the angle lies.
a. sin 𝜃 > 0 and cos 𝜃 > 0
b. sin 𝜃 > 0 and cos 𝜃 < 0
c. sec 𝑡 > 0 and tan 𝑡 > 0
(You try) d. sin 𝛼 > 0 and tan 𝛼 < 0
34
Example 6: Find the values of the six trigonometric functions using the functional value and constraint.
4
3𝜋
5
2
a. cos 𝜃 = − , 𝜋 < 𝜃 <
b. tan 𝑡 =
5
12
,
3𝜋
2
< 𝑡 < 2𝜋
35
c. cos 𝛼 = −
c. csc 𝜃 = 4,
4
13
, tan 𝛼 > 0
cot 𝜃 < 0
36
Example 7: Evaluate without a calculator the sin, cos, and tan of the angles. It might help to sketch a
picture of the angle and draw in the reference angle.
a. 330°
𝜋
c. − 2
e.
−
b. -405°
d.
11𝜋
6
f.
10𝜋
4
15𝜋
3
37
Example 8: Find two values of 𝜃 that satisfy the equation. Give your answers in degrees and radians.
Do not use a calculator.
a. cos 𝜃 = −
√2
2
b. sin 𝜃 =
√3
2
1
c. tan 𝜃 = −√3
d. cos 𝜃 = 2
e. sec 𝜃 = 2
f. cot 𝜃 = 1
38
How does a reference angle help you determine the value of any trig function of any angle?
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
____________________________________________________________________________________
39
Homework for 4-4
Pg 285 37-43 odd
Pg 284 1,5,7,13,15, 17-23 odd, 67-73 odd
Pg 285 29-35 odd, 53-65 odd
Pg 285 91- 95 odd
40
Continue homework for 4-4 on this extra page
Pg 285 37-43 odd
Pg 284 1,5,7,13,15, 17-23 odd, 67-73 odd
Pg 285 29-35 odd, 53-65 odd
Pg 285 91- 95 odd
41
Continue homework for 4-4 on this extra page
Pg 285 37-43 odd
Pg 284 1,5,7,13,15, 17-23 odd, 67-73 odd
Pg 285 29-35 odd, 53-65 odd
Pg 285 91- 95 odd
42
43