Barnett/Ziegler/Byleen College Algebra with Trigonometry, 6 th Edition Chapter Seven Trigonometric Identities & Conditional Equations Copyright © 1999 by the McGraw-Hill Companies, Inc.

Basic Trigonometric 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. Suggested Steps in Verifying Identities

Sum Identities sin( x + y ) = sin x cos y + x sin y cos( x + y ) =cos x y – sin x sin y tan( x + y ) = tan x + y 1 – x y Difference Identities sin( x – y ) = sin x cos y – x sin y cos( x – y ) =cos x y + sin x sin y tan( x – y ) = tan x – y 1 + x 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

Double and Half-Angle Identities

Product-Sum Identities Sum-Product Identities

y = cosx x y 1 –1 y = 0.5 –4  2  –2  4  cos x = 0.5 has infinitely many solutions for –  < x <  y = cosx x y 1 –  cos x = 0.5 has two solutions for 0 < x < 2  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. Some Suggestions for Solving Trigonometric Equations