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Chapter 4 Trigonometry Section 4 Trigonometric Functions of Any Angle to infinity and beyond . . . ! Math Analysis 1 Objectives Evaluate trigonometric functions of any angle to ± infinity Use reference angles to evaluate trigonometric functions. Use trig models to solve real world problems 2 Definitions of trigonometric functions 3 Find the trig functions using a point on the terminal side syr cxr tyx 4 Quadrant angles x θ y cos θ sin θ 0 π/2 π 3π/2 2π 5 Trigonometric function signs Star Sin/csc Trig Tan/cot All All 6 trig functions are positive in quadrant 1 Class Cos/sec 6 Find the quadrant 1. sinθ > 0; cosθ <0 2. sinθ < 0; cosθ > 0 7 Reference angle Let θ be an angle in standard position. Its reference angle is the acute angle θ‘ formed by the terminal side of θ and the horizontal axis. 8 Reference angles are always measured between the x axis and the terminal side of the angle (always +!) Notice the butterfly shape 9 Drawing Reference Angles Find the reference angle θ' , and sketch θ and θ' in standard position. 1) θ = -145° 2) 11 3 10 Finding the Trig functions of any tan135 angle 1. 2. Find the reference angle ref 180 135 45 Determine the value of the given trig function of tan 45 1 the reference angle. 135 QII 3. Determine the sign— tan is in Q II based on the quadrant of the given (original) angle tan135 1 11 Now: Find all 6 trig functions of an angle Find the six trigonometric functions of θ with the given constraint. 15 tan ; 8 sin 0 12 Now, let’s make it tougher The terminal side of θ lies on the line y = (1/3)x in quadrant I. Find the values of the 6 trig functions of θ by finding a point on the line. sin θ = cos θ = csc θ = sec θ = tan θ = cot θ = 13 Try These: Find the values of the six trig functions of θ with the given constraints. sin θ = 0; where sec θ = -1 Find the reference angle θ‘ and sketch it and the angle θ in standard position. a) θ = 3.5 b) θ = 750° 14 Find the trig functions of an angle Evaluate the sine, cosine, and tangent of the angle without using a calculator. -405° 15 Do 88(a)&(c) 16 Do 91 17 Objectives Evaluate trigonometric functions of any angle to ± infinity Use reference angles to evaluate trigonometric functions. 1. 2. 3. Syr Cxr Tyx Quadrantal Angles Find ref < Find trig value of ref < Determine sign of trig value for given < Use trig models to solve real world problems Draw, identify, solve 18 Pg. 284: 3-52 (multiples of 3-- i.e. 3, 6, 9, etc.) 19