Barnett/Ziegler/ByleenCollege Algebra with Trigonometry, 6th Edition

Chapter Seven

Trigonometric Identities & Conditional Equations

Copyright © 1999 by the McGraw-Hill Companies, Inc.

Reciprocal Identities

csc x = 1

sin x sec x = 1

cos x cot x = 1

tan x

Quotient Identities

tan x = sin xcos x cot x =

cos xsin x

Identities for Negatives

sin(–x) = –sin x cos(–x) = cos x tan(–x) = –tan x

Pythagorean Identities

sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x

Basic Trigonometric Identities

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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.

Suggested Steps in Verifying Identities

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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 + y) = tan x + tan 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 – y) = tan x – tan y

1 + tan x tan y

Cofunction Identities

Replace 2 with 90° if x is in degrees.

cos

2 – x = sin x sin

2 – x = cos x tan

2 – x = cot x

Sum, Difference, and Cofunction Identities

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Double-Angle Identities

sin 2x = 2 sin x cos x

cos 2x = cos2 x – sin2 x = 1 – 2 sin2 x = 2 cos2 x – 1

tan 2x = 2 tan x

1 – tan2 x =

2 cot xcot2 x – 1

= 2

cot x – tan x

Half-Angle Identities

sinx2 = ±

1 – cos x2

cosx2 = ±

1 + cos x2

tanx2 = ±

1 – cos x1 + cos x =

sin x1 + cos x =

1 – cos xsin x

where the sign is determined by the quadrant in which x2 lies.

Double and Half-Angle Identities

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sin x cos y = 12 [sin (x + y ) + sin (x – y )]

cos x sin y = 12 [sin (x + y ) – sin (x – y )]

sin x sin y = 12 [cos(x – y ) – cos(x + y )]

cos x cos y = 12 [cos(x + y ) + cos(x – y )]

Product-Sum Identities

sin x + sin y = 2 sin x + y

2 cos x – y

2

sin x – sin y = 2 cos x + y

2 sin x – y

2

cos x + cos y = 2 cos x + y

2 cos x – y

2

cos x – cos y = –2 sin x + y

2 sin x – y

2

Sum-Product Identities

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y = cos x

x

y

1

–1

y = 0.5

–4 2–2 4

cos x = 0.5 has infinitely many solutions for – < x <

y = cos x

x

y

1

–1

0.5

2

cos x = 0.5 has two solutions for 0 < x < 2

Trigonometric Equations

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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.

Some Suggestions for Solving Trigonometric Equations

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