Precalculus Chapter 5
Johnson and Prange
Trigonometric Identities
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Learning Target
5.1.1
I can use the Reciprocal, Quotient, and the
Pythagorean Identities to evaluate functions.
I can use the Reciprocal, Quotient, and the
5.1.2 Pythagorean Identities to simplify and rewrite
expressions.
5.2.1 I can verify trigonometric identities.
5.3.1
I can solve trigonometric equations on the interval
[0,2π).
Practice for the
Learning Target
Score on
Learning
Target Quiz
pg 345 1-13 odd,
skip #9
pg 345 27 – 37
odd skip #35
51, 55,
61 -71 odd
Pg 346 39,41,43
Pg 353 1-9 odd,
Pg 364 7-29 odd
Essential Questions for the chapter
1. How are the six trigonometric and circular functions related to each other?
Essential Questions for the course
1. How is this similar or different from what I have done before?
2. What can I do to retain what I have learned?
1. Does my answer make sense? If not, what do I do?
2. Do I need help, and where do I go to find it?
3. How would a calculator make this problem easier to do?
4. How do I explain or justify my work to myself and others?
5. What is the given information and how do I use it?
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LEARNING TARGET QUIZ SCORING RUBRIC
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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.
 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.
1
MINIMAL UNDERSTANDING
I wasn’t sure which mathematical operations to use, and my plan didn’t work.
 I tried several things, but didn’t get anywhere.
0
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.
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Reciprocal Identities
Quotient Identities
MEMORIZE
sin x 
1
csc x
cos x 
1
sec x
tan x 
1
cot x
csc x 
1
sin x
sec x 
1
cos x
cot x 
1
tan x
Pythagorean Identities
sin 2 x  cos 2 x  1
tan x 
sin x
cos x
MEMORIZE
cot x 
cos x
sin x
MEMORIZE
1  tan 2 x  sec 2 x
1  cot 2 x  csc 2 x
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Date _______
Notes: 5-1
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can use the Reciprocal, Quotient, and the Pythagorean Identities to evaluate functions.
Vocabulary
Reciprocal
Identity
Quotient
Identity
Pythagorean
Identity
sin and csc
cos and sec
tan and cot
Example 1 Use the given values to evaluate the six trigonometric functions.
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A.
sec 𝜃 = −
C.
cot 𝑢 = −5 and sin 𝑢 =
2
and tan 𝜃 < 0
B.
csc 𝜃 = 2 and tan 𝜃 =
√3
3
√26
26
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HW pg 345 1-13 odd, skip #9
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5-1 Warm Up(s)
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Date _______
Notes: 5-1
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can use the Reciprocal, Quotient, and the Pythagorean Identities to simplify and rewrite
expressions.
Vocabulary
Reciprocal
Identity
common
denominator
Quotient
Identity
Pythagorean
Identity
sin and csc
cos and sec
tan and cot
conjugate
factor
GCF
cross multiply
like terms
State the 3 Pythagorean Identities:
1.
2.
3.
Example 1 Simplify using trig identities
A.
cos  tan 
B.
csc 
sec 
C.
cos 2 x(sec2 x  1)
D.
sec 2   1
sin 2 
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E.
1
tan x  1
2
F.
tan 𝜃
cot 𝜃
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5-1 HW pg 345 27 - 37 odd skip #35
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5-1 Warm Up(s)
1. Factor (𝑥 2 − 4)
2.
3.
Factor 𝑥 4 − 24
5.
Find the conjugate of the following expressions.
A. 5 + 2𝑖
Factor 𝑥 4 − 2𝑥 2 + 1
4. Solve for sin 2 𝑥 given sin 2 𝑥 + cos 2 𝑥 = 1
B. −4𝑖
C. 1 + sin 2 𝑥
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Date _______
Notes: 5-1
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can use the Reciprocal, Quotient, and the Pythagorean Identities to simplify and rewrite
expressions.
Vocabulary
Reciprocal
Identity
common
denominator
Quotient
Identity
Pythagorean
Identity
sin and csc
conjugate
factor
GCF
State the 3 Pythagorean Identities:
1.
cos and sec
cross
multiply
tan and cot
like terms
2.
3.
Example 1 Multiply – then simplify using trig identities
A.
(cot x  csc x)(cot x  csc x)
B.
(3  3sin x)(3  3sin x)
B.
sec 2 x  1
sec x  1
Example 2 Factor - then simplify using trig identities
A.
sec2 x tan 2 x  sec2 x
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C.
1  2 cos 2 x  cos 4 x
D.
sec4 x  tan 4 x
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5-1 HW pg 346 51, 55, 61, 63
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5-1 Warm Up(s)
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Date _______
Notes: 5-1
Essential Questions:
2. How are the six trigonometric and circular functions related to each other?
Learning Targets:
2. I can use the Reciprocal, Quotient, and the Pythagorean Identities to simplify and rewrite
expressions.
Vocabulary
Reciprocal
Identity
common
denominator
Quotient
Identity
Pythagorean
Identity
sin and csc
conjugate
factor
GCF
State the 3 Pythagorean Identities:
1.
cos and sec
cross
multiply
tan and cot
like terms
2.
3.
Example 1 Add or Subtract - then simplify
1
1
A.
+ cos 𝜃(sin 𝜃)
sin 𝜃
B.
1
1

sec x  1 sec x  1
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Example 2 Rewrite the expression so that it is not in fraction form.
A.
5
tan x  sec x
B.
tan 2 x
csc x  1
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5-1 HW pg 346 65, 67, 69, 71
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HW 5-1 Mixed Practice
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Continue HW 5-1 Mixed Practice
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5-2 Warm Up(s)
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Date _______
Notes: 5-2
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can verify trigonometric identities.
Vocabulary
Reciprocal Identity
common denominator
Quotient Identity
cross multiply
Pythagorean Identity
factor
verify
GCF
conjugate
like terms
What are the 5 guidelines for Verifying Trigonometric Identities?
1._____________________________________________________________________________
2.____________________________________________________________________________
3.____________________________________________________________________________
4.____________________________________________________________________________
5.____________________________________________________________________________
Example 1 Verify each identity.
A.
sec2   1
 sin 2 
2
sec 
B. cos2   sin 2   1  2sin 2  (Tip: Use identities.)
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C.
1
1

 2sec 2 x (Tip: Add fractions and use identities.)
1  sin x 1  sin x
D.
 tan
2
x  1 cos 2 x  1   tan 2 x (Tip: Use identities before multiplying.)
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E. tan x  cot x  sec x csc x (Tip: Convert to sines and cosines.)
F.
tan x cot x
 sec x (Tip: Convert to sines and cosines.)
cos x
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5-2 HW pg 353 1-9 odd
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Continue 5-2 HW pg 353 1-9 odd
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5-2 Warm Up(s)
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Date _______
Notes: 5-2
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can verify trigonometric identities.
Vocabulary
Reciprocal Identity
common denominator
Quotient Identity
cross multiply
Pythagorean Identity
factor
verify
GCF
conjugate
like terms
What are the 5 guidelines for Verifying Trigonometric Identities?
1._____________________________________________________________________________
2.____________________________________________________________________________
3.____________________________________________________________________________
4.____________________________________________________________________________
5.____________________________________________________________________________
A.
cos x
 sec x  tan x (Tip: Multiply by denominator’s conjugate, use an identity, and separate fractions.)
1  sin x
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B.
sin 
1  cos 

(Tip: Cross multiply)
1  cos 
sin 
C. tan 4 x  tan 2 x sec 2 x  tan 2 x (Tip: Take out a GCF.)
D. tan 5 x  tan 3 x sec2 x  tan 3 x (Tip: Take out a GCF.)
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HW 5-2 pg 346 39,41,43
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HW 5-1 and 5-2 Review for Trigonometric Identities
Verify the identity algebraically.
1
 sin x cos x
tan x  cot x
1. sec4   tan 4   1  2 tan 2 
2.
2
2
2
2. sin x  cos x tan x  0
4. (csc x  cot x)(csc x  cot x)  1
5.
sec x  cos x
 tan 2 x
cos x
6. cos 2   csc2   sin 2   cot 2 
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7.
 sec2 x 
1
sin x(sin x  csc x)
8.
tan x  1
 sec x
sin x  cos x
9. 2sec 2 x  2sec 2 x sin 2 x  sin 2 x  cos 2 x  1
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5-3 Warm Up(s)
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Date _______
Notes: 5-3
Essential Questions:
1. How are the six trigonometric and circular functions related to each other?
Learning Targets:
1. I can solve trigonometric equations on the interval [0,2π).
Vocabulary
factor
unit circle
GCF
sin and csc
like terms
cos and sec
interval notation
tan and cot
domain
Example 1 Solve each equation on the interval [0, 2 ). You need the unit circle.
A. Solve cos x 
1
[0, 2 )
2
C. Solve cos x 
1
for all 
2
B. Solve tan x  1
[0, 2 )
D. Solve tan x  1 for all 
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Find all solutions of the equations on the interval [0, 2 ).
Example 2: Isolate the trigonometric function.
2sin x 1  0
You Try:
1  2cos x  0
Example 3: Collect Like Terms
sin x  2   sin x
You Try:
sin x 1   sin x
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Example 4: Using Square Roots
3 tan 2 x  1
You Try:
csc 2 x  2  0
Example 5: Factoring
cot x cos 2 x  2 cot x
You Try:
sec x csc x  csc x
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Example 6: Factoring
2sin 2 x  sin x  1  0
You Try:
2 cos 2 x  cos x  1  0
Example 7: Rewrite as a single trigonometric function
2sin 2 x  3cos x  3  0
You Try:
2 cos 2 x  3sin x  3  0
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5-3 HW
After examples 1 – 4 Pg 364 7, 9, 11, 17
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5-3 HW
After examples 5 and 6 Pg 364 21, 23
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5-3 HW
After example 7 Pg 364 13, 15, 19, 27, 29
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5.1 - 5.3 Quiz Review
You may use a graphing calculator..
5-1-1 I can use the Reciprocal, Quotient, and the Pythagorean Identities to evaluate functions.
Part I: 1 Use the given values to evaluate the six trigonometric functions.
3
1. csc 𝜃 = − 2 and tan 𝜃 < 0
2.
sec 𝜃 = 2 and tan 𝜃 =
√3
3
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5-1-2 I can use the Reciprocal, Quotient, and the Pythagorean Identities to simplify and rewrite
expressions.
Part II: Matching
Simplify each trigonometric expression. Write the letter of the answer in each blank.

______3. 1  cos x
2
  csc x 
a. sec x
2
c. cos x
e.
 tan x
g. csc x
b.
1
d. 1
f. sin x
h.
tan x
______4. sec x cos x
2
i. sin x
2
k. sec x
______5.
j. sin x tan x
l. sec x  tan x
2
2
tan x csc x
sec2 x  1
______6.
sin 2 x
______7. cot x  csc x
2
2
______8. sin x sec x
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5-2-1 I can verify trigonometric identities.
Part III - Verify: Your work must be detailed and legible.
9. sec x  sec x cos x  tan x
2
10.
2
2
2
1
1

 2csc2 
1  cos  1  cos 
(Tip: GCF, Identities, Sines and Cosines)
(Tip: Common Denominator)
11. sin x  cos x cot x  csc x (Tip: Sines and Cosines, Common Denominator, Identities)
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5-3-1 I can solve trigonometric equations on the interval [0,2π).
Part IV: Find all solutions in the interval [0,2 ) . Your solutions must be written in
radians.
12.
2sin x  3  0
12._________________
13.
csc2 x  csc x  2
13._________________
14.
sin 2 x  cos x  1
14._________________
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