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Trigonometric Identities M 120 Precalculus V. J. Motto Preliminary Comments • Remember an identity is an equation that is true for all defined values of a variable • We are going to use the identities that we have already established to "prove" or establish other identities. Let's summarize the basic identities we have. Right Triangle Definitions Unit Circle Definitions Basic Trigonometric Identities • Reciprocal Identities Basic Trigonometric Identities • Quotient Identities Basic Trigonometric Identities • Pythagorean Identities Basic Trigonometric Identities • Even-Odd Identities Establish the following identity: sin csc cos sin 2 2 Let's sub in here using reciprocal identity sin csc cos sin 1 2 2 sin cos sin sin 2 We are done! We've shown the LHS equals the RHS 2 1 cos sin 2 2 sin sin 2 2 We often use the Pythagorean Identities solved for either sin2 or cos2. sin2 + cos2 = 1 solved for sin2 is 1 - cos2 which is our left-hand side so we can substitute. In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match. sin Establish the following identity: csc cot Let's sub in here using reciprocal identity and quotient identity 1 cos sin We worked on csc cot 1 cos LHS and then RHS but never 1 cos sin moved things sin 1 cos FOIL denominator across the = sign sin 1 cos sin sin 1 cos combine fractions sin cos 1 cos 11cos Another trick if the 1 cos sin 1 cos denominator is two terms 2 with one term a 1 and the sin 1 cos other a sine or cosine, multiply top and bottom of 1 cos sin 1 cos the fraction by the conjugate 2 sin sin and then you'll be able to use the Pythagorean Identity on the bottom 1 cos 1 cos sin sin Hints for Establishing Identities • Find common denominators when there are fractions. • Squared functions often suggest Pythagorean Identities. • Work on the more complex side first. • A denominator of 1 + trig function suggest multiplying top & bottom by conjugate which leads to the use of Pythagorean Identity. • When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities. • Trigonometric Identities are like puzzles! They are fun and test you algebra skills and insights. • Enjoy them! Attitude does make a difference in success. Other Trigonometric Identities • Identities expressing trigonometric function in terms of their complements. Other Trigonometric Identities • Sum formulas of sine and cosine The derivation involves the use of geometry. Other Trigonometric Identities • Double angle formulas for sine and cosine These are easily derived from the previous identities.