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on
13-
1
NAME DATE PERIOD
Co
pyr
ight
© G
lenc
oe/
McG
raw
-Hill
, a d
ivis
ion
of
The
McG
raw
-Hill
Co
mp
anie
s, In
c.
PDF Pass
Chapter 13 5 Glencoe Algebra 2
13-1 Study Guide and InterventionTrigonometric Identities
Find Trigonometric Values A trigonometric identity is an equation involving trigonometric functions that is true for all values for which every expression in the equation is defined.
Basic
Trigonometric
Identities
Quotient Identities tan θ = sin θ −
cos θ cot θ = cos θ
−
sin θ
Reciprocal Identities csc θ = 1 −
sin θ sec θ = 1
−
cos θ cot θ = 1
−
tan θ
Pythagorean Identities cos2 θ + sin2 θ = 1 tan2 θ + 1 = sec2 θ cot2 θ+ 1 = csc2 θ
Find the exact value of cot θ if csc θ = - 11 −
5 and 180° < θ < 270°.
cot2 θ + 1 = csc2 θ Trigonometric identity
cot 2 θ + 1 = (-
11 −
5 )
2 Substitute - 11
−
5 for csc θ.
cot 2 θ + 1 = 121 −
25 Square - 11
−
5 .
cot 2 θ = 96 −
25 Subtract 1 from each side.
cot θ = ±
4 √ � 6
−
5
Take the square root of each side.
Since θ is in the third quadrant, cot θ is positive. Thus, cot θ = 4 √ � 6 −
5 .
ExercisesFind the exact value of each expression if 0° < θ < 90°.
1. If cot θ = 4, find tan θ. 2. If cos θ = √ � 3
−
2 , find csc θ.
3. If sin θ = 3 −
5 , find cos θ. 4. If sin θ = 1 −
3 , find sec θ.
5. If tan θ = 4 −
3 , find cos θ. 6. If sin θ = 3 −
7 , find tan θ.
Find the exact value of each expression if 90° < θ < 180°.
7. If cos θ = -
7 −
8 , find sec θ. 8. If csc θ = 12 −
5 , find cot θ.
Find the exact value of each expression if 270° < θ < 360°.
9. If cos θ = 6 −
7 , find sin θ. 10. If csc θ = -
9 −
4 , find sin θ.
Example
1 −
4
4 −
5
3 −
5
-
8 −
7
-
√ �� 13 −
7
2
3 √ � 2
−
4
3 √ �� 10
−
20
-
√ �� 119 −
5
-
4 −
9
001_020_ALG2_A_CRM_C13_CR_660516.indd 5001_020_ALG2_A_CRM_C13_CR_660516.indd 5 12/21/10 11:57 AM12/21/10 11:57 AM
NAME DATE PERIOD
Co
pyrig
ht © G
lencoe/M
cGraw
-Hill, a d
ivision o
f The M
cGraw
-Hill C
om
panies, Inc.
PDF Pass
Chapter 13 6 Glencoe Algebra 2
13-1 Study Guide and Intervention (continued)
Trigonometric Identities
Simplify Expressions The simplified form of a trigonometric expression is written as a numerical value or in terms of a single trigonometric function, if possible. Any of the trigonometric identities can be used to simplify expressions containing trigonometric functions.
Simplify (1 - cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ.
(1 - cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ = sin 2 θ · 1 −
cos θ · cos θ
−
sin θ + sin θ
−
cos θ · 1 −
cos θ · cos 2 θ
= sin θ + sin θ
= 2 sin θ
Simplify sec θ · cot θ −
1 - sin θ - csc θ
−
1 + sin θ .
sec θ · cot θ −
1 - sin θ - csc θ
−
1 + sin θ =
1 −
cos θ · cos θ
−
sin θ
−
1 + sin θ -
1 −
sin θ
−
1 + sin θ
= 1 −
sin θ (1 + sin θ) - 1 −
sin θ (1 - sin θ)
−−
(1 - sin θ)(1 + sin θ)
= 1 −
sin θ + 1 - 1 −
sin θ + 1 −−
1 - sin 2 θ
= 2 −
cos 2 θ or 2 sec2 θ
Exercises
Simplify each expression.
1. tan θ · csc θ −
sec θ 2. sin θ · cot θ
−
sec 2 θ - tan 2 θ
3. sin 2 θ - cot θ · tan θ −−
cot θ · sin θ 4. cos θ
−
sec θ - tan θ
5. tan θ · cos θ −
sin θ + cot θ · sin θ · tan θ · csc θ 6. csc 2 θ - cot 2 θ −
tan θ · cos θ
7. 3 tan θ · cot θ + 4 sin θ · csc θ + 2 cos θ · sec θ 8. 1 - cos 2 θ −
tan θ · sin θ
Example 1
Example 2
-cos θ
cos θ1
2
9
1 + sin θ
csc θ
cos θ
001_020_ALG2_A_CRM_C13_CR_660516.indd 6001_020_ALG2_A_CRM_C13_CR_660516.indd 6 12/21/10 11:57 AM12/21/10 11:57 AM
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