1 Chapter 4 Trigonometry Section 4 Trigonometric Functions of Any Angle to infinity and beyond... ! Math Analysis

2 Objectives  Evaluate trigonometric functions of any angle to ± infinity  Use reference angles to evaluate trigonometric functions.  Use trig models to solve real world problems

3 Definitions of trigonometric functions Syr Cxr Tyx

4 Find the trig functions using a point on the terminal side syr cxr tyx (-3, 4)

5 Quadrant angles θcos θsin θ 0 π/2 π 3π/2 2π2π xy

6 Trigonometric function signs A ll S tar T rig C lass Sin/csc Tan/cotCos/sec All 6 trig functions are positive in quadrant 1

7 Find the quadrant 1. sin θ > 0; cos θ <0 2. sin θ 0

8 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.

 Reference angles are always measured between the x axis and the terminal side of the angle ( always +!)  Notice the butterfly shape 9

10 Drawing Reference Angles Find the reference angle θ', and sketch θ and θ' in standard position. 1) θ = -145° 2)

11 Finding the Trig functions of any angle 1. Find the reference angle 2. Determine the value of the given trig function of the reference angle. 3. Determine the sign— based on the quadrant of the given (original) angle

12 Now: Find all 6 trig functions of an angle Find the six trigonometric functions of θ with the given constraint.

13 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 =

14 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 = 3.5b) θ = 750°

15 Find the trig functions of an angle Evaluate the sine, cosine, and tangent of the angle without using a calculator. -405°

16 Do 88(a)&(c)

17 Do 91

18 Objectives  Evaluate trigonometric functions of any angle to ± infinity Syr Cxr Tyx Quadrantal Angles  Use reference angles to evaluate trigonometric functions. 1. Find ref < 2. Find trig value of ref < 3. Determine sign of trig value for given <  Use trig models to solve real world problems Draw, identify, solve

19 Pg. 284: 3-52 (multiples of 3-- i.e. 3, 6, 9, etc.)