REVIEW WARM-UP : With a partner try and solve the following 10 questions.
(Hint: Think about what you know about right triangles)
4
5
I. If sin A = , and A is in quadrant I, find each of the following (Answer should be a ratio)
1. cos A
2. tan A
3. csc A
II. Solve. Assume 𝜃 is between 0o and 90o
4. If tan 𝜃 = 3, find cot 𝜃.
5. If sin 𝜃 =
5
,
13
find cos 𝜃.
2
6. If cos 𝜃 = 3 , find tan 𝜃.
6. If csc 𝜃 = 5, find sec 𝜃.
III. Express each value as a function of an angle in quadrant 1. (All in degrees)
8. sin 400
9. tan 475
10. cos 220
REVIEW WARM-UP : With a partner try and solve the following 10 questions.
(Hint: Think about what you know about right triangles)
4
I. If sin A = 5, and A is in quadrant I, find each of the following (Answer should be a ratio)
1. cos A
2. tan A
3. csc A
II. Solve. Assume 𝜃 is between 0o and 90o
4. If tan 𝜃 = 3, find cot 𝜃.
5. If sin 𝜃 =
5
,
13
find cos 𝜃.
2
6. If cos 𝜃 = 3 , find tan 𝜃.
6. If csc 𝜃 = 5, find sec 𝜃.
III. Express each value as a function of an angle in quadrant 1. (All in degrees)
8. sin 400
9. tan 475
10. cos 220
Pre-Calculus/Trigonometry
Unit 7
Name:_______________________________
Date:__________________ Block:______
5.1 Trigonometric Identities
Question to be answered:
What are the trig identities and how can they be used to simplify expressions?
There are three main types of trigonometric identities we will study.
_______________________, _______________________,
___________
Type I: Reciprocal Identities
sin Θ =
cos Θ =
tan Θ =
csc Θ =
sec Θ =
cot Θ =
Type II: Quotient Identities
Given
Rewrite with
opp, adj and
hyp.
sinθ
cosθ
Simplify
Fraction
Rewrite as
one trig.
function
tan Θ =
cot Θ =
Practice:
1. sin θ =
3
1
and cosθ=− , find all the ratio 6 trig functions without a triangle.
2
2
2. sec x = 2 and sin x=
2
, find all the ratio 6 trig functions without a triangle.
2
Type III: Pythagorean Identities
r = ____________
Pythagorean Theorem:
_____ + _____ = ____
Rewrite using x, y, and r:
Rewrite using trig:
(**1st Identity)
(________ , _________)
Unit Circle
Divide by sin2Θ :
Divide 1st identity by cos2 Θ:
Rewrite these identities solving for each term of each equation.
Verifying Trigonometric Identities: To verify a trigonometric identity, you transform one
side of the equation into the same form as the other side, by using basic trig identities and
properties of algebra.
Tips:
1) ____________________________________________________________________
2) Look for opportunities to use:
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
________________________________________________________________
3) ____________________________________________________________________
4) ____________________________________________________________________
There are a variety of strategies that can be used to simply trigonometric identities.
1. Substitute Basic Identities
2. Look for Pythagorean Substitutions and
Variations
3. Look for algebraic manipulation
(i.e. factoring, distributing, FOIL, etc.)
4. When you multiply by reciprocal function it
will always get you one
5. Splitting fractions will help (sometimes).
Note: Whole Denominator Goes to Both
6. Multiply by “one” to create a common
denominator.
7. Change everything to sin and cos
equivalents.
***REMINDER***
ONLY EVER WORK ON
ONE SIDE OF THE
IDENTITY!!!!!
Pre-Calculus/Trigonometry
Unit 7
Name:_______________________________
Date:__________________ Block:______
Directions: Verify each trigonometric identity.
1)
tan x csc x
1
sec x
2) tan x cos 2 x  sin x cos x
3)
tan x
 tan 2 x
cot x
4) cos x tan x csc x  1
csc x
 sin x
2
1  cot x
1
cos 2 x
5)

1
sin 2 x sin 2 x
6)
7) 1  sin x1  sin x  cos 2 x
8) sin x  cos x tan x  2 sin x
9) cos x  sec x  sin x tan x
10) 1  tan 2 xcos 2 x  1
Pre-Calculus/Trig
Name ____________________________
Verifying Identities Worksheet
Date ________________ Block _______
Directions: Verify each trigonometric identity. Complete all work on a separate piece of paper.
1)
cos3 θ + sin2 θ cos θ = cos θ
2)
3)
sec θ sin θ = tan θ
sec2 θ−1
= tan θ
tan θ
4)
8)
9)
sec θ sin θ cot θ =1
cos2 θ − sin2 θ =2 cos2 θ −1
11)
cot θ sin θ = cos θ
12)
13)
sin θ 1+ csc θ = sin θ +1
1+ tan2 θ cos2 θ =1
14)
5)
7)
15)
17)
19)
21)
23)
sec θ + tan θ sec θ − tan θ =1
1−2 csc θ
= tan θ −2 sec θ
cot θ
sin θ + cos θ cot θ = csc θ
cos θ
= sec θ
1− sin2 θ
31)
cot θ
= sin θ cos θ
1+ cot2 θ
sin θ+ cos θ
= sec θ + csc θ
sin θ cos θ
2
1+ sin θ
1+ sin θ
=
1− sin θ
cos2 θ
csc θ cos2 θ + sin θ = csc θ
33)
sin θ
35)
csc θ − sin θ = cot θ cos θ
25)
27)
29)
37)
39)
41)
43)
45)
cot θ
+ csc θ = cos2 θ +1
sec θ
1+ tan θ
=1+ cot θ
tan θ
csc4 θ − cot4 θ =2 csc2 θ −1
1+ sec θ
= csc θ
tan θ+ sin θ
1
= sec θ + tan θ
sec θ− tan θ
1− sin θ = tan θ cot θ − cos θ
6)
10)
16)
18)
20)
22)
24)
26)
28)
30)
32)
34)
36)
38)
40)
csc2 θ − cos2 θ csc2 θ =1
csc θ
= cot θ
sec θ
cot θ
= tan θ
csc2 θ−1
cot θ csc θ tan2 θ = sec θ
cos2 θ − sin2 θ =1−2 sin2 θ
tan θ
= sin θ
sec θ
1+ tan θ 2 = sec2 θ +2 tan θ
cos θ = sec θ − sin θ tan θ
sec θ
= sec θ − cos θ
csc2 θ
sec2 θ−1
= tan θ
tan θ
cos θ csc θ − sec θ = cot θ −1
tan2 θ − tan2 θ sin2 θ = sin2 θ
1+ tan2 θ
= sec4 θ
cos2 θ
sec θ+ tan θ
= sin θ sec2 θ
cos θ+ cot θ
1+ sec θ
= csc θ
tan θ+ sin θ
csc2 θ
= sec2 θ
csc2 θ−1
2 cos2 θ− sin2 θ+1
=3 cos θ
cos θ
1
1
+
=2 csc2 θ
1− cos θ 1+ cos θ
cos θ+ tan θ
= sec θ + cot θ
sin θ
cos θ+ cot θ
= cos θ
csc θ+1
42)
2− sec2 θ 1−2 sin2 θ
=
sec θ
cos θ
44)
1+ cos θ = cot θ sin θ + tan θ
2)
1− sin θ
sec θ − tan θ 2 =
1+ sin θ
Extra Practice
1)
3)
5)
7)
tan θ
= sin θ cos θ
1+ tan2 θ
1− cos θ 1− cos θ
=
1+ cos θ
sin θ
sin4 θ − cos4 θ =1−2 cos2 θ
sin θ − cos θ 2 +2 sin θ − cos θ = 1+2 sin θ (1− cos θ )
4)
6)
sec θ+ tan θ 1+ sin θ
=
sec θ− tan θ cos θ
csc2 θ−1
sin2 θ
− cot2 θ = sin2 θ
cos2 θ