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November 20, 2014 Chapter 5 Analytic Trigonometry Section 5.1 Using Fundamental Identities Objectives: Use fundamental trig. identities to evaluate trig. functions and simplify trigonometric expressions. In this chapter we will use identities to... 1. Evaluate trig. functions, 2. Simplify trig. expressions, 3. Develop additional trig. identities, 4. Solve trig. equations Review of Identities covered so far: Reciprocal Identities sin u = csc u = 1 csc u 1 sin u cos u = 1 sec u tan u = 1 cot u sec u = 1 cos u cot u = 1 tan u Quotient Identities sin u u cot u = cos sin u tan u = cos u Pythagorean Identities sin2u + cos2u = 1 tan2u + 1 = sec2u Cofunction Identities sin (90˚ - u) = cos u sec (90˚ - u) = csc u cos (90˚ - u) = sin u csc (90˚ - u) = sec u 1 + cot2u = csc2u cot (90˚ - u) = tan u tan (90˚ - u) = cot u Even Even/Odd Odd Identities sin (-u) = - sin u Odd tan (-u) = - tan u Odd sec (-u) = sec u Even cos (-u) = cos u Even cot (-u) = - cot u Odd csc (-u) = - csc u Odd Tip: For each one of these, there are 2 other versions that you should be familiar with. November 20, 2014 Ex1: If csc u = -5/3 and cos u > 0, find the values of the other five trigonometric functions. sin u = -3/5 cos u = 4/5 tan u = -3/4 s c Ex2: Find the value of tan u given that sec u = 2. tan u = √3 Note: Could use Pyt 1 + tan2u = Ex3: Simplify. a) sec x - tan x sin x b) csc2x cot x - cot x c) tan x sin x + cos x d) sec t tan t 1 + sec t tan t a) cosx b) cot3x c) sec x d) cot t November 20, 2014 Ex4: Factor the following trig. expressions. a) cos2x - 1 a) (cos x +1)(cos x b) sin2u - 3 sin u - 10 b) (sin u +2)(sin u c) sec2t - tan t - 3 c) Ex5: Rewrite sec 1x - 1 so that it is not a fraction. (tan t + 1)(tan Ex5: cot x csc x + co Ex6: Use the substitution x = 3sin u, 0 < u < π/2, to express √9 - x as a function of u. 2 Ex6: Ex7: Is sec x sin3x + sin x cos x = tan x an identity? Yes o r Yesno!? ify... h To ver e grap h t k s c f value 1. Che o e l b y a raicall ck a t b e e h g l C a . 2 lent equiva w o h 3. S 3 cos u