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This is a movie about Trigonometry

This is a movie about Trigonometry. C3 Reciprocal and Inverse Trig Functions Directed by J Wathall and her Year13 A level Maths class. Reciprocal functions. What is the reciprocal of y = 3x + 3 ? Yes it is :

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This is a movie about Trigonometry

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  1. This is a movie aboutTrigonometry 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 1000 , cosec 2600 and cot( 4/3) c to 3 sig figs. • Volunteer: WAC evaluate the exact value of cot 1350, sec 2250 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 • 3) Using a different coloured pen now sketch y = sec x

  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 = 900, 2700 etc • What is the period of the curve? (one full cycle) • 3600

  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= 1800, 3600, etc • What is the period of the curve? (one full cycle) • 3600

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

  16. Y = cot x • Write down when the asymptotes occur. • X=0,1800, etc • What is the period of the curve? (one full cycle) • 1800

  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 27th 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 • Sinxsecx • Sinxcosx(secx+cosecx)

  19. Showing: volunteer • Cotx cosecx = cos3 x Sec2 x+ cosec2x Q 1,2,3 and 4 Ex 6C

  20. Showing Melody • Cotx cosecx = cos3 x Sec2 x+ cosec2x

  21. Q4f Show that

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

  23. Showing • Cotx cosecx = cos3 x Sec2 x+ cosec2x

  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 2x. • Divide this equation by sin2 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:

  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 • 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 4cosec2 x -9 = cot x for 0≤ x ≤ 360

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