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Pre β Calculus Unit 4 Section 5.1 Notes β Trigonometric Identities Objectives: Identify basic trig identities - Use basic trig identities to - find trig values - simplify and rewrite expressions Identities An equation is an identity if the left side is equal to the right side for all values of the variable for which both sides are defined. IDENTITY π₯ 2 β9 π₯β3 NON-IDENTITY =π₯+3 sinx = 1 β cosx Example 1: 3 a) If cos ο± = , find sec ο± 4 5 3 4 4 b) If sec x = and tan x = , find sinx Pythagorean Identities Recall from 4.3 that trigonometric functions can be defined on a unit circle as shown. Notice that for any angle, sine and cosine are directed lengths of the legs of a right triangle with hypotenuse 1. Apply the Pythagorean Theorem to this right triangle to establish another basic trig identity. (-sin ο±)2 + (-cos ο±)2 = 12 sin2 ο± + cos2 ο± = 12 - Other forms of the Pythagorean Identities Example 2: a) If cot ΞΈ = 2 and cos ΞΈ < 0, find sin ΞΈ and cos ΞΈ. b) Find the value of cscο± and cotο± if tanο± = β 4 3 and cosο± < 0 Simplfying β’ Helpful Hints β No fractions β 1 trig function β If you see addition or subtraction with a 1 and/or a (trig function)2 will most likely use P.T. Identities β Usually donβt rewrite sin or cos as their reciprocals, but will write tan, csc, sec, and cot β Factor out a GCF β Common Denominators / Conjugate Example 4: Use identities to simplify a) 1 cos π₯ (1 β sin2 π₯) b) cscx β cosx cotx Example 5: Use identities to simplify a) cos x ο tan x β sin x ο cos 2 x b) cos x ο sin2x β cosx Example 6: Use identities to simplify a) sec π₯ 1βsec π₯ β sec π₯ 1+sec π₯ b) 1+cos π₯ sin π₯ Example 7: Rewrite as an expression that does not involve fractions. a) c) 1+tan2 π₯ csc2 π₯ csc π₯ 1βsin π₯ b) sin2 π₯ 1+cos π₯ + sin π₯ 1+cos π₯ Match the trigonometric identity with one of the expressions: 1. sec x cos x a) sec x 2. tan x csc x b) β1 3. cot 2 π₯ β csc 2 π₯ c) 1 4. (1 β cos 2 π₯)(csc π₯) d) sin x PRACTICE: 1. sin ΞΈ sec ΞΈ cot ΞΈ 2. cot x sec x sin x 3. tan x csc x cos x 4. tan π cot π csc π