1. This is a movie about Trigonometry C3 Reciprocal and Inverse Trig Functions Directed by J Wathall and her Year13 A level Maths class

2. Reciprocal functions What is the reciprocal of y = 3x + 3 ? Yes it is : The reciprocal means ONE OVER the function. Or in a fraction it means to change the denominator and numerator

3. Inverse functions The inverse of a function maps the output of a function back to the input. THIS IS NOT THE RECIPROCAL! For example the function y = 3x + 3 has an inverse of Notice the inverse is not the same as the reciprocal. The inverse is NOT one over!

4. Reciprocal Trig Functions What is the reciprocal of cos x? What is the reciprocal of sin x? What is the reciprocal of tan x? We have special names for these reciprocal functions.

5. Here they are… Here we must remember that the denominator cannot equal zero so cos x, sin x and tan x are not defined for the value zero.

6. Example 1 Volunteer : Using your calculator evaluate sec 100 0, cosec 260 0 and cot( 4  /3) c to 3 sig figs. cot( 4  /3) c to 3 sig figs. Volunteer: WAC evaluate the exact value of cot 135 0, sec 225 0 and cot( 4  /3) c

7. What do the reciprocal graphs look like? 1) Complete this table for y = sec x: 2) Sketch the curve y = cos x for -180 < x < 180 for -180 < x < 180 3) Using a different coloured pen now sketch y = sec x x030456070808595100110120135150180210 y

8. A review of last lesson Do you remember how to sketch the reciprocal trig functions? Sketch y= cos x and on the same curve sketch y= sec x for -180<x<180 labeling all asymptotes

9. Tada!

10. Or on a larger scale y= secx

11. Facts about y = sec x Write down when the asymptotes occur. X = 90 0, 270 0 etc What is the period of the curve? (one full cycle) 360 0

12. What is the difference between the graphs of y = sinx and y = cos x? Yes you are correct. So the y = cosec x curve is exactly the same as the y = sec x curve but a shift to the right by 90 0. Can you sketch this on your graph paper using another colour. Don’t forget to draw your asymptotes

13. Y = cosec x- blue curve

14. Facts about y = cosec x Write down when the asymptotes occur. X= 180 0, 360 0, etc What is the period of the curve? (one full cycle) 360 0

15. Lastly y = cot x Write down three facts about this curve.

16. Y = cot x Write down when the asymptotes occur. X=0,180 0, etc What is the period of the curve? (one full cycle) 180 0

17. Transformations of the Reciprocal Trig Functions. Let us use Autograph to help us understand these transformations. See worksheet work through guided examples. Homework Monday 27 th Aug: If you want an A All of ex 6A, 6B If you want a B every other question in 6A,6B for Wednesday

18. Simplifying Trig expressions Examples Simppppplify SinxsecxSinxcosx(secx+cosecx)

19. Showing: volunteer Cotx cosecx = cos 3 x Cotx cosecx = cos 3 x Sec 2 x+ cosec 2 x Sec 2 x+ cosec 2 x Q 1,2,3 and 4 Ex 6C

20. Showing Melody Cotx cosecx = cos 3 x Cotx cosecx = cos 3 x Sec 2 x+ cosec 2 x Sec 2 x+ cosec 2 x

21. Q4f Show that

22. Homework help! Is this a quadratic? Ex 6H

23. Showing Cotx cosecx = cos 3 x Cotx cosecx = cos 3 x Sec 2 x+ cosec 2 x Sec 2 x+ cosec 2 x

24. Ex 6c q6H A quadratic in disguise

25. Solving trig equations Sec x = -2.5 for the interval 0<x<360 Cot 2x = 0.6 for the interval 0<x<360 Ex 6C 5,6,7

26. Solving Gillean, Jocelyn Secx = -2.5 Cot 2x = 0.6

27. Homework 6C

28. 6C q7D

29. Another form of an Identity Starting with the identity Divide this equation by cos 2 x. Divide this equation by sin 2 x.

30. Two new identities

31. Lots of examples If tan x = -5/12 and x is obtuse find the exact value of A) sec x B) sin x Use a RAT

32. More examples Prove

33. One more interesting one

34. Ex 6D more practice

35. The Inverse Trig Functions Remember an inverse means a function which maps the output back to the input and the graph is a reflection about the line y = x. So we do not confuse the reciprocal trig functions we use a special notation for the inverse trig functions. The are called arcsinx, arccosx and arctanx.

36. Some conditions For an inverse function to exist the function must be a one to one mapping. We restrict the domain of y = sin x, y = cos x and y = tan x for the inverse to exist. Let us use Autograph again to help us see what arcsinx, arccosx and arctanx looks like.

37. One to one mapping y = cos x

38. y = arccosx Here the domain is -1<x<1 The range is 0<y< 

39. Y=arccosx You must remember here that the domain is restricted to 0 ≤ x ≤  So if we were simplifying We would only look at the second quadrant Why?

40. Example Simplify the following This is the same as: This is the same as:

41. Y = sin x Go to www.mathsnet.net for beautiful applet

42. Y = arcsin x Domain -1<x<1 Range -  /2<y<  /2

43. Domain Here for y = arcsinx the domain is -  /2 ≤ x ≤  /2 -  /2 ≤ x ≤  /2 So to simplify a problem like this: We only look at fourth quadrant why?

44. Y = arctan x

45. Domain of y = arctanx You can see x is real so the domain is The range is So simplifying We find

46. Inverse trig applets Click here Inverse trig graphs as a reflection

47. Example Click here for worked examples:

48. Ex 6E Q6b

49. Mixed exercise 6F Proving identities

50. Using trig identities Solve the equation 4cosec 2 x -9 = cot x for 0≤ x ≤ 360

51. The End The mind map Click here Click here