1. Chapter Seven Barnett/Ziegler/Byleen
College Algebra with Trigonometry, 7th Edition
Chapter Seven
Trigonometric Identities & Conditional Equations
Copyright © 2001 by the McGraw-Hill Companies, Inc.

2. Basic Trigonometric Identities
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3. Suggested Steps in Verifying Identities
1. Start with the more complicated side of the identity,
and transform it into the simpler side.
2. Try algebraic operations such as multiplying, factoring,
combining fractions, splitting fractions, and so on.
3. If other steps fail, express each function in terms of sine and
cosine functions, and then perform appropriate algebraic
operations.
4. At each step, keep the other side of the identity in mind.
This often reveals what you should do in order to get there.
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4. Sum, Difference, and Cofunction Identities
Sum Identities
sin( x + y
) = sin x
cos y +
cos x
sin y
cos( x + y
) =
cos x
cos y
– sin x
sin y
tan x +
tan y
tan( x + y
) = 1 –
tan x
tan y
Difference Identities
sin( x – y
) = sin x
cos y –
cos x
sin y
cos( x – y
) =
cos x
cos y
+ sin x
sin y
tan x –
tan y
tan(
x – y
) = 1 +
tan x
tan y
Cofunction Identities æ  ö ç
Replace
with
90° if x is in
degrees. ÷ 2 è ø æ  ö æ  ö æ  ö
cos ç – x ÷
= sin x
sin ç – x ÷ =
cos x
tan ç – x ÷
= cot x 2 2 2 è ø è ø è ø
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5. Double and Half-Angle Identities
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6. Product-Sum Identities
Sum-Product Identities
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7. Trigonometric Equationsy y
= cos x 1 y
= 0.5 x –4  –2  2  4  –1
cos x = 0.5 has infinitely many solutions for –  < x <  y y
= cos x 1
0.5 x 2 
cos x = 0.5 has two solutions for 0 < x < 2 –1
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8. Some Suggestions for Solving Trigonometric Equations
1. Regard one particular trigonometric function as a variable,
and solve for it.
2. Consider using algebraic manipulation such as factoring.
3. Consider using identities.
4. After solving for a trigonometric function, solve for the variable
following the procedures discussed in the preceding section.
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