1. Graphs of other Trig Functions Section 4.6

2. Cosecant Curve What is the cosecant x? Where is cosecant not defined? ◦Any place that the Sin x = 0 The curve will not pass through these points on the x-axis. x = 0, π, 2 π

3. Cosecant Curve Drawing the cosecant curve 1) Draw the reciprocal curve 2) Add vertical asymptotes wherever curve goes through horizontal axis 3) “Hills” become “Valleys” and “Valleys” become “Hills”

4. Cosecant Curve y = Csc x→ y = Sin x 1

5. Cosecant Curve y = 3 Csc (4x – π)→ y = 3 Sin (4x – π) a = 3b = 4 Per. = dis. = c = π P.S. = -3 3

6. Cosecant Curve y = -2 Csc 4x + 2→ y = -2 Sin 4x + 2 2 4

7. Secant Curve What is the secant x? Where is secant not defined? ◦Any place that the Cos x = 0 The curve will not pass through these points on the x-axis.

8. Secant Curve y = Sec 2x→ y = Cos 2x 1

9. Secant Curve y = Sec x→ y = Cos x 1

10. Graph these curves 1) y = 3 Csc (πx – 2π) 2) y = 2 Sec (x + ) 3) y = ½ Csc (x - ) 4) y = -2 Sec (4x + 2π)

11. y = 3Csc (πx – 2π)→ y = 3 Sin (π x – 2π) -3 3

12. y = 2Sec (x + )→ y = 2 Cos (x + ) -2 2

13. y = ½ Csc (x - )→ y = ½ Csc (x - ) - ½ ½

14. y = -2 Sec (4π x + 2 π) -2 Cos (4π x + 2 π) -2 2

15. Graph of Tangent and Cotangent Still section 4.6

16. Tangent Define tangent in terms of sine and cosine Where is tangent undefined?

17. y = Tan x

18. Tangent Curve So far, we have the curve and 3 key points Last two key points come from the midpoints between our asymptotes and the midpoint ◦Between and 0 and between and 0 → and

19. y = Tan x x und. 0 0 1 1

20. For variations of the tangent curve 1) Asymptotes are found by using: A1. bx – c = A2. bx – c = 2) Midpt. = 3) Key Pts: and

21. y = 2Tan 2x x und. bx – c = 2x= x =

22. y = 2Tan 2x x und. 0 0 -22 Midpt = K.P. = = = 0

23. y = 4Tan x und. 0 0 -44

24. y = 4Tan x und. 0 0 -44

25. Cotangent Curve Cotangent curve is very similar to the tangent curve. Only difference is asymptotes bx – c = 0bx – c = π → 0 and π are where Cot is undefined

26. y = 2Cot x und. 0 π 2-2 2Cot

27. x und. 0 π 2-2 y = 2Cot 2Cot

28. x und. 03-3 y = 3 Cot 3Cot

29. Graph the following curves: y = 2 Cos ( + ) + 2 y = 2 Sin ( + π ) + 1 y = 5 Tan (4x – π )

30. y = 2 Cos ( + ) + 2 a = 2b = Per. = dis. = c = P.S. = 2 4 d =

31. y = 2 Sin ( + π ) + 1 a = 2b = Per. = dis. = c = P.S. = 1 3 d =

32. y = 5Tan (4x – π) 5Tan (4x – π) x und. 0-55

33. Graph the following curves: y = -3 Sec (x + ) y = -2 Csc (x - ) y = ½ Cot (x – )

34. y = -3 Sec (x + ) -3 Sec ( x + ) -3 3

35. y = -2 Csc (x - )→ y = -2 Csc (x - ) - 2 2

36. x und. 0½- ½ y = ½ Cot ½ Cot