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Barnett/Ziegler/Byleen College Algebra with Trigonometry, 6th Edition Chapter Seven Trigonometric Identities & Conditional Equations Copyright © 1999 by the McGraw-Hill Companies, Inc. Basic Trigonometric Identities Reciprocal Identities 1 csc x = sin x 1 sec x = cos x 1 cot x = tan x Quotient Identities sin x tan x = cos x cos x cot x = sin x Identities for Negatives sin(–x) = –sin x cos(–x) = cos x tan(–x) = –tan x tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x Pythagorean Identities sin2 x + cos2 x = 1 7-1-78 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. 7-1-79 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 tan x 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 tan 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 7-2-80 Double-Angle Identities Double and Half-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 2 cot x 2 = = 1 – tan2 x cot2 x – 1 cot x – tan x Half-Angle Identities x sin 2 = ± 1 – cos x 2 x cos 2 = ± 1 + cos x 2 x tan 2 = ± 1 – cos x sin x 1 – cos x = = 1 + cos x 1 + cos x sin x x where the sign is determined by the quadrant in which 2 lies. 7-3-81 Product-Sum Identities 1 sin x cos y = 2 [sin (x + y ) + sin (x – y )] 1 cos x sin y = 2 [sin (x + y ) – sin (x – y )] 1 sin x sin y = 2 [cos(x – y ) – cos(x + y )] 1 cos x cos y = 2 [cos(x + y ) + cos(x – y )] Sum-Product Identities sin x + sin y = 2 sin x+y x–y cos 2 2 sin x – sin y = 2 cos x+y x–y sin 2 2 cos x + cos y = 2 cos x+y x–y cos 2 2 cos x – cos y = –2 sin x+y x–y sin 2 2 7-4-82 Trigonometric Equations y y = cos x 1 y = 0.5 –4 –2 2 4 x –1 cos x = 0.5 has infinitely many solutions for – < x < y y = cos x 1 0.5 2 –1 x cos x = 0.5 has two solutions for 0 < x < 2 7-5-83 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. 7-5-84