Download 13-1 Study Guide and Intervention

NAME
13-1
DATE
PERIOD
Study Guide and Intervention
Trigonometric 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.
sin θ
tan θ = −
cos θ
cot θ = −
Reciprocal Identities
1
csc θ = −
1
sec θ = −
Pythagorean Identities
cos2 θ + sin2 θ = 1
tan2 θ + 1 = sec2 θ
cos θ
sin θ
sin θ
cos θ
1
cot θ = −
tan θ
cot2 θ+ 1 = csc2 θ
11
Find the exact value of cot θ if csc θ = - −
and 180° < θ < 270°.
Example
5
cot2 θ + 1 = csc2 θ
5)
(
11
cot 2 θ + 1 = - −
Trigonometric identity
2
121
cot θ + 1 = −
25
96
2
cot θ = −
25
4 √
6
cot θ = ± −
5
2
11
Substitute - −
for csc θ.
5
11
Square - −
.
5
Subtract 1 from each side.
Take the square root of each side.
4 √6
5
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
Since θ is in the third quadrant, cot θ is positive. Thus, cot θ = − .
Exercises
Find the exact value of each expression if 0° < θ < 90°.
√3
2
1. If cot θ = 4, find tan θ.
2. If cos θ = − , find csc θ.
3
3. If sin θ = −
, find cos θ.
1
4. If sin θ = −
, find sec θ.
4
5. If tan θ = −
, find cos θ.
3
6. If sin θ = −
, find tan θ.
5
3
3
7
Find the exact value of each expression if 90° < θ < 180°.
7
7. If cos θ = - −
, find sec θ.
8
12
8. If csc θ = −
, find cot θ.
5
Find the exact value of each expression if 270° < θ < 360°.
6
9. If cos θ = −
, find sin θ.
7
Chapter 13
9
10. If csc θ = - −
, find sin θ.
4
5
Glencoe Algebra 2
Lesson 13-1
Basic
Trigonometric
Identities
Quotient Identities
NAME
DATE
13-1
PERIOD
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.
Example 1
Simplify (1 - cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ.
1
(1 - cos2 θ) sec θ cot θ + tan θ sec θ cos2 θ = sin 2 θ · −
cos θ
cos θ
sin θ · 1 ·
−
− cos 2 θ
+−
·
cos θ
sin θ
cos θ
= sin θ + sin θ
= 2 sin θ
Example 2
θ · cot θ
csc θ
−
- −
.
Simplify sec
1 - sin θ
1 + sin θ
cos θ
1
−
·−
cos θ sin θ
1
−
sec θ · cot θ
csc θ
sin θ
− - − = − - −
1 - sin θ
1 + sin θ
1 + sin θ
1 + sin θ
1
1
−
(1 + sin θ) - −
(1 - sin θ)
θ
sin θ
−−
= sin
(1 - sin θ)(1 + sin θ)
1
1
−
+1-−
+1
θ
sin θ
−−
= sin
2
Copyright © Glencoe/McGraw-Hill, a division of The McGraw-Hill Companies, Inc.
1 - sin θ
2
= −
or 2 sec2 θ
cos 2 θ
Exercises
Simplify each expression.
θ · csc θ
−
1. tan
sin θ · cot θ
2. −
2
2
2
θ - cot θ · tan θ
−−
3. sin
cos θ
4. −
θ · cos θ
−
5. tan
+ cot θ · sin θ · tan θ · csc θ
2
θ - cot 2 θ
−
6. csc
7. 3 tan θ · cot θ + 4 sin θ · csc θ + 2 cos θ · sec θ
1 - cos θ
8. −
sec θ
sec θ - tan θ
sec θ - tan θ
cot θ · sin θ
tan θ · cos θ
sin θ
Chapter 13
6
2
tan θ · sin θ
Glencoe Algebra 2