Report

Chapter 7 Trigonometric Identities and Equations 7.1 BASIC TRIGONOMETRIC IDENTITIES Reciprocal Identities 1 csc 1 csc = sin 1 tan = cot sin = cos = 1 sec 1 sec = cos 1 cot = tan These identities are derived in this manner sin = and csc = which gives you sin = Quotient Identities cos sin = tan = cot If using a unit circle as reference, these identities were derived using cos = = tan Pythagorean Identities sin² + cos² = 1 tan² + 1 = sec² 1 + cot² = csc² Opposite Angle Identities sin [-A] = -sin A cos [-A] = cos A 7.2 VERIFYING TRIGONOMETRIC IDENTITIES Tips For Verifying Trig Identities • Simplify the complicated side of the equation • Use your basic trig identities to substitute parts of the equation • Factor/Multiply to simplify expressions • Try multiplying expressions by another expression equal to 1 • REMEMBER to express all trig functions in terms of SINE AND COSINE 7.3 SUM AND DIFFERENCE IDENTITIES Difference Identity for Cosine Cos (a – b) = cosacosb + sinasinb • As illustrated by the textbook, the difference identity is derived by using the Law of Cosines and the distance formula Sum Identity for Cosine Cos (a + b) = cos (a - (-b)) The sum identity is found by replacing -b with b *Note* If a and b represent the measures of 2 angles then the following identities apply: cos (a ± b) = cosacosb ± sinasinb Sum/Difference Identity For Sine sinacosb + cosasinb = sin(a + b) – sum identity for sine If you replace b with (-b) you can get the difference identity of sine. sin (a – b) = sinacosb - cosasinb Sum & Difference Tan[a ± b] = ± 1 ± This identity is used as both the sum and difference identity.