TRIGONOMETRIC FUNCTIONS OF ANY ANGLE

1. TRIGONOMETRIC FUNCTIONS OF ANY ANGLE Section 4.4

2. Objectives: • Evaluate trigonometric functions of any angle. • Use reference angles to evaluate trigonometric functions. • To evaluate trigonometric functions of real numbers.

3. Unit Circle Rationale Recall that when using the unit circle to evaluate the value of a trig function, cosθ = x and sin θ = y. What we didn’t point out is that since the radius (hypotenuse) is 1, the trig values are really cosθ = x/1 and sin θ = y/1. So what if the radius (hypotenuse) is not 1?

4. Exercise 1 Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ.

5. Exercise 1 Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cos θ, and tan θ. Realize that the triangle formed by the point (4, 3) is similar to a triangle in the unit circle. To get to that unit circle triangle, we would have to scale down the larger triangle by dividing by the scale factor. In this case, that’s 5, the length of the larger hypotenuse.

6. Exercise 1 Let θ be an angle whose terminal side contains the point (4, 3). Find sin θ, cosθ, and tan θ. Solution:

7. Trig of Any Angle Let (x, y) be a point on the terminal side of an angle θ in standard position with

8. Exercise 2 The previous definitions imply that tan θ and sec θ are not defined when x = 0. So what values of θ are we talking about? They also imply that cot θ and cscθ are not defined when y = 0. So what values of θ are we talking about?

9. Exercise 3 Let θ be an angle whose terminal side contains the point (−2, 5). Find the six trig functions for θ.

10. Signs of Trig Functions We can find the sign of a particular trig function based on which quadrant (x, y) lies within.

11. Exercise 4 Given sin θ = 4/5 and tan θ < 0, find cosθ and cscθ. 5 4 -3

12. Reference Angles When building the unit circle, for 120° we drew a triangle with the x-axis to form a 60° angle. This 60° angle was the reference angle for 120°.

13. Reference Angles If we connect the trig of any angle to right triangle trigonometry, we need a reference angle that is acute to be able to evaluate the function.

14. Reference Angles Let θ be an angle in standard position. It’s reference angle is the acute angle θ’ formed by the terminal side of θ and the x-axis.

15. Reference Angles Let θ be an angle in standard position. It’s reference angle is the acute angle θ’ formed by the terminal side of θ and the x-axis.

16. Exercise 5 Find the reference angle for each of the following. • 213° • 1.7 rad • −144°

17. Reference Angles When your angle is negative or is greater than one revolution, to find the reference angle, first find the positive coterminal angle between 0° and 360° or 0 and 2π.

18. Reference Angles How Reference Angles Work: Same except maybe a difference of sign.

19. Reference Angles To find the value of a trig function of any angle: Find the trig value for the associated reference angle Pick the correct sign depending on where the terminal side lies

20. Exercise 6 Evaluate: • sin 5π/3 • cos (−60°) • tan 11π/6

21. Exercise 7 Let θ be an angle in Quadrant III such that sin θ = −5/13. Find a) sec θ and b) tan θ using trig identities.

22. 4.4: Trig of Any Angle Objectives: • Evaluate trigonometric functions of any angle. • Use reference angles to evaluate trigonometric functions. • To evaluate trigonometric functions of real numbers. Assignment: • Pg. 294-296: #1 – 23 odd, 29 – 75 odd