Copyright © 2000 by the McGraw-Hill Companies, Inc. Barnett/Ziegler/Byleen Precalculus: A Graphing Approach Chapter Six Trigonometric Identities & Conditional Equations

Copyright © 2000 by the McGraw-Hill Companies, Inc. Basic Trigonometric Identities

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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

Copyright © 2000 by the McGraw-Hill Companies, Inc. Double-Angle and Half-Angle Identities

Product-Sum Identities Sum-Product Identities Copyright © 2000 by the McGraw-Hill Companies, Inc

Copyright © 2000 by the McGraw-Hill Companies, Inc. 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

Copyright © 2000 by the McGraw-Hill Companies, Inc. 1. Regard one particular trigonometric function as a variable, and solve for it. (a)Consider using algebraic manipulation such as factoring, combining or separating fractions, and so on. (b)Consider using identities. 2. After solving for a trigonometric function, solve for the variable. Suggestions for Solving Trigonometric Equations Algebraically