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Verifying Trig Identities (5.1) JMerrill, 2009 (contributions from DDillon) Trig Identities Identity: an equation that is true for all values of the variable for which the expressions are defined Ex: sin x tan x cos x or (x + 2) = x + 2 Conditional Equation: only true for some of the values Ex: tan x = 0 or x2 + 3x + 2 = 0 Recall y sin r x cos r y tan x r csc y r sec x x cot y Recall - Identities Reciprocal Identities sin 1 csc 1 cos sec 1 tan cot Also true: 1 csc sin 1 sec cos 1 cot tan Recall - Identities Quotient Identities sin tan cos cos cot sin Fundamental Trigonometric Identities Negative Identities (even/odd) sin sin csc csc cos cos sec sec tan tan cot cot These are the only even functions! Recall - Identities Cofunction Identities sin cos 2 cos sin 2 tan cot 2 cot tan 2 sec csc 2 csc sec 2 Recall - Identities Pythagorean Identities sin cos 1 2 2 1 cot csc 2 2 tan 1 sec 2 2 Simplifying Trig Expressions • Strategies • Change all functions to sine and cosine (or at least into the same function) • Substitute using Pythagorean Identities • Combine terms into a single fraction with a common denominator • Split up one term into 2 fractions • Multiply by a trig expression equal to 1 • Factor out a common factor Simplifying # 1 cot x sin x cos x sin x sin x cos x sin x sin x cos x 2 Simplifying #2 cos x sin x sin x cos 2 x sin 2 x sin x sin x 2 2 cos x sin x sin x 1 sin x csc x Simplifying #3 1 cos x 2 cos x 2 2 sin x 2 cos x tan x 2 Simplifying #4 cos x sin x tan x sin x cos x sin x cos x 2 sin x cos x cos x 2 2 cos x sin x cos x cos x 1 cos x sec x Simplifying #5 2 sin x cos x cos x 2 3 sin x sin x cos x 3 2 sin x sin x cos x 2 sin x 2 Proof Strategies • Never cross over the equal sign (you cannot assume equality) • Transform the more complicated side of the identity into the simpler side. • Substitute using Pythagorean identities. • Look for opportunities to factor • Combine terms into a single fraction with a common denominator, or split up a single term into 2 different fractions • Multiply by a trig expression equal to 1. • Change all functions to sines and cosines, if the above ideas don’t work. ALWAYS TRY SOMETHING!!! Example sin cos csc 1 cos sin • Prove • 2 fractions that need to be added: • Shortcut: sin sin cos 1 cos 1 cos sin sin2 cos cos2 1 cos sin 1 cos 1 cos sin 1 csc sin Show cos x 1 cot x cot x 2 2 2 cos 2 x 1 cot 2 x 1 + cot2x = csc2 x cos x csc x 1 csc x 2 sin x 2 2 2 1 cos x 2 sin x 2 cos 2 x sin 2 x cot x 2 tan x cot x Prove tan x 2 csc x tan x cot x csc 2 x sin x cos x cos x sin x csc 2 x sin x cos x cos x sin x 2 csc x sin 2 x cos 2 x sin x cos x csc2 x 1 sin x cos x 1 sin 2 x sin 2 x 1 sin x cos x 1 sin x cos x tan x