1. Section 4.4

2. In first section, we calculated trig functions for acute angles. In this section, we are going to extend these basic definitions to cover any angle. θ θ

3. Plot the point (-3,4) Label the hypotenuse r and find its length. r = 5 5 -3 4 θSin θ = Cos θ = Tan θ =

4. Definitions of Trig Functions of Any Angle Let θ be an angle in standard position with (x,y) a point on the terminal side. Then: Csc θ = Sec θ = Cot θ = Sin θ = Cos θ = Tan θ =

5. Find the 6 trig functions of θ given that the ray ends at the point (-15, -8) -15 -8 17 Csc θ = Sec θ = Cot θ = Sin θ = Cos θ = Tan θ =

6. Find the 6 trig functions of θ given that the ray ends at the point (12, -5) 12 -5 13 Csc θ = Sec θ = Cot θ = Sin θ = Cos θ = Tan θ =

7. Quadrants In which quadrants was the Sine positive? I and II In which quadrants was the Cosine positive? I and IV In which quadrants was the Tangent positive? I and III

8. Quadrants All Trig Functions are positive Sine is positive Cosine is positive Tangent is positive All Students TakeCalculus

9. What quadrant is θ in if: a) Sin θ > 0 and Cos θ < 0 b) Tan θ > 0 and Cos θ < 0 c) Sin θ < 0 and Tan θ < 0 d) Cos θ > 0 and Tan θ > 0 → II → III → IV → I

10. Given that Tan θ = - and Sin θ > 0, find the remaining 5 trig functions of θ. What quadrant?II -24 7 25 Csc θ = Sec θ = Cot θ = Sin θ = Cos θ = Tan θ =

11. Given that Cos θ = - and Sin θ < 0, find the remaining 5 trig functions of θ. What quadrant?III -4 5 -3 Csc θ = Sec θ = Cot θ = Sin θ = Cos θ = Tan θ =

12. Given that Sin θ = - and Tan θ < 0, find the remaining 5 trig functions of θ. What quadrant?IV -15 17 8 Csc θ = Sec θ = Cot θ = Sin θ = Cos θ = Tan θ =

13. What did we learn How to find the trig functions of an angle given a point on its terminal side How to determine the quadrant of an angle based on trig functions How to find the trig functions based on one function and criteria Homework: Page 297, 1-24 odd

14. Find the Sin, Cos, and Tan trig functions of θ given that the ray ends at the point (5,0) 5 y = 0 Sin θ = Cos θ = Tan θ =

15. Quadrant Angles On our Cartesian plane, we have 5 critical points: Find the Sine of these 5 angles Sin 0 = 0 Sin =1 Sin π = 0 Sin = Sin 2π = 0

16. Graph of the Sine Curve Using these 5 points, we can create the Sine Curve 0

17. Quadrant Angles Using the same process, find the Cos of the 5 critical points. Cos 0 = 1 Cos =0 Cos π = Cos = 0 Cos 2π = 1

18. Graph of the Cosine Curve Using these 5 points, we can create the Sine Curve 0

19. Reference Angles The acute angle formed by the terminal side of an angle and the horizontal axis. For an angle θ, we use θ’ to denote the reference angle

20. Reference Angles What is the reference angle for 210º Where is there an acute angle between the terminal side of the angle and the horizontal axis? θ’ = 210 – 180 = 30º

21. Reference Angles Find the reference angles for the following: a) 330 º b) 225º c) -225º d) 750º = 360 º - 330 º = 30 º = 225 º - 180 º = 45 º = -180 º - -225 º = 45 º = 750 º - 720 º = 30 º

22. Reference Angles In general, for any angle θ θ’ = θ θ’ = 180 - θ θ’ = π - θ θ’ = θ - 180 θ’ = θ - π θ’ = 360 - θ θ’ = 2π - θ

23. Reference Angles Find the reference angle for 2 nd Quadrant: → π – θ = π – =

24. Reference Angles So far, all we have been finding are reference angles. We use reference angles to find the exact value of angles that are not acute. We will use this for the remainder of the year. “GTK” – Good to Know

25. Finding the Exact Value 1. Find the reference angle 2. Find the trig function of the reference angle 3. Check the sign of the function

26. Sin 200 º 1. Find the reference angle 2. Find the Sin of the reference angle 3. Is it positive or negative?

27. Cos 330 º 1. Find the reference angle 2. Find the Sin of the reference angle 3. Is it positive or negative?

28. Find the Sin, Cos, and Tan of 135 º Reference Angle = Quadrant = Sin 135º = Cos 135º = Tan 135º =

29. Find the Sin, Cos, and Tan of -240 º Reference Angle = Quadrant = Sin -240º = Cos -240º = Tan -240º =

30. Find the Sin, Cos, and Tan of Reference Angle = Quadrant = Sin = Cos = Tan =

31. Find the: a) Sin b) Csc c) Tan d) Csc e) Cot