More with Angles Reference Angles

Transcription

1 More with Angles Reference Angles A reference angle is the angle formed by the terminal side of an angle θ, and the (closest) x axis. A reference angle, θ', is always 0 o <θ'<90 o. 23

2 Example Sketch the angle. Then find its reference angle. A. 280 o B. 24

3 The Unit Circle The Unit Circle is a circle of radius 1 unit. Since angles do not change measure and we going to be looking at ratios, it doesn't matter how big the circle is. So, we use the most basic circle we can to talk about trigonometric functions. 25

4 The Unit Circle In Degrees: 26

5 The Unit Circle In Radians: 27

6 The Unit Circle Together: 28

7 Things You Should Know from Before: 60 o 45 o 30 o 45 o 29

8 Things You Should Know from Before For any angle, θ, in a triangle: sinθ = cosθ = tanθ = sine cosine tangent 30

9 FINDING TRIG VALUES FOR STANDARD ANGLES Find the sine, cosine and tangent of 30 o. 31

10 FINDING TRIG VALUES FOR STANDARD ANGLES Find the sine, cosine and tangent of 225 o. 32

11 FINDING TRIG VALUES FOR STANDARD ANGLES Find the sine, cosine and tangent of 2π/3 radians. 33

12 Determine the values of the three basic trigonometric functions for the following: o 2. π radians 3. 5π/ π/3 34

13 The OTHER Trigonometric Functions We know the three basic trig functions from "Soh Cah Toa." But, there are three other trigonometric functions called the "reciprocal functions." Recall, The reciprocal functions are the cotangent, secant, and cosecant functions. 35

14 Another Version of the Tangent Function Remember: But, and So... 36

15 What is the sign of the sine (and the other trigonometric functions)? Some things to remember: 1) sine goes with y 2) cosine goes with x 3) tangent uses a combination of x and y sinθ cosθ tanθ sinθ cosθ tanθ sinθ cosθ tanθ sinθ cosθ tanθ 37

16 Trigonometric Values of Special (common) Angles 120 o sin(120 o ) = cos(120 o ) = tan(120 o ) = 38

17 TRIG VALUES FOR ANY ANGLE (on any circle) (x, y) Find all six trig functions for the angle,θ, created by the point (x, y). r θ 39

18 Rotational Trigonometry Trigonometric Functions of Any Angle Let θ be any angle in standard position and the point P(x, y) be a point on the terminal side of θ. Let r represent the nonzero distance from the origin to P. Then, And, the trigonmetric functions of θ are: x y 40

19 Example 1 Find the value of all six trig functions for the angle formed by the terminal side that passes through the point (3, 4). 41

20 Example 2 Let the point ( 4, 5) be a point on the terminal side of the angle θ, in standard position. Find the values of all six trigonometric functions. 42

21 Example 3 Let P be a point on the terminal side of the angle θ, in Quadrant III, such that. Find the values of the other trigonometric functions. 43

22 Example 4 Find the value of all six trig functions for the angle formed by the terminal side that passes through the point ( 5, 6). 44

23 Example 5 Let, where sinθ <0. Find the exact values of the five remaining trigonometric functions of θ. 45

24 The Trigonometric Values of the Quadrantal Angles Points on the quadrantal angles. 46

25 Example 3 Evaluate each of the following trigonometric functions. If not defined, write "undefined." A) sin 180 o B) tan 3π C) sec 90 o D) cot 2π 47

26 Assignment 1. Create/Practice making your own unit circle. 2. Book Assignment (on calendar) (soon you will begin having timed mini quizzes) 48