1. Use the formula Area = 1/2bcsinA Think about the yellow area What’s sin (α+β)?

2. Sin(A+B) Ξ ?

3. Sin(x+30) = ?

4. Sin(A+B) Ξ ? Sin(x+30) = ? Cos(A-B) Ξ ?

5. Sin(A+B) Ξ ? Sin(x+30) = ? Cos(A-B) Ξ ? Cos(x-60) = ?

6. Sin(A+B) Ξ ? Sin(x+30) = ? Cos(A-B) Ξ ? Cos(x-60) = ? Tan(A+B) Ξ ?

7. Sin(A+B) Ξ ? Sin(x+30) = ? Cos(A-B) Ξ ? Cos(x-60) = ? Tan(A+B) Ξ ? Tan(A+60) = ?

8. Use the formula Area = 1/2bcsinA Think about the yellow area What’s sin (α+β)?

9. Trig addition formulae Aims: To learn the trig addition formula. To solve equations and prove trigonometrical identities using the addition formulae.

10. Trig Addition Formulae Does ? and So, We cannot simplify the brackets as we do in algebra because they don’t mean multiply. l.h.s. = r.h.s. =

11. Trig Addition Formulae The addition formulae are in your formulae booklets, but they are written as: Notice that the cos formulae have opposite signs on the 2 sides. Use both top signs in a formula or both bottom signs.

12. Trig Addition Formulae Using the Addition Formulae e.g. Prove the following: Proof: l.h.s. ( formulae (1) and (2) )

13. Trig Addition Formulae Have a go Relay race In groups of 3-4

14. Exam questions

20. Plenary Step by step on whiteboards 1.Expand sin(x+30) and cos(x+60) 2.Simplify by using exact values 3.Look at the form you’re aiming for and rearrange 4.Use rearranged form and rearrange again to make single trig ratio 5.Find principal value and any symmetry or periodicity values

22. Double Angle Formulae Objectives: To recognise and learn the double angle formulae for Sin 2A, Cos 2A and Tan 2A. To apply the double angle formulae to solving trig equations and proving trig identities.

23. Double Angle Identities (1) sin (A + A) = sin A cos A + cos A sin A sin (2A) sin (2A) = 2 sin A cos A sin (A + B) = sin A cos B + cos A sin B Setting A = B

24. Double Angles Identities (2) cos (A + A) = cos A cos A - sin A sin A cos (2A) cos (2A) = cos 2 A - sin 2 A Since, sin 2 x + cos 2 x = 1 cos (2A) = cos 2 A – (1 - cos 2 x) = 2 cos 2 A - 1 cos (A + B) = cos A cos B - sin A sin B Setting A = B cos (2A) = 1 - sin 2 x – sin 2 A = 1 - 2 sin 2 A

25. SUMMARY The double angle formulae are:

26. N.B. The formulae link any angle with double the angle. For example, they can be used for and We use them to solve equations to prove other identities to integrate some functions and

27. Activity: Trig double angle Card match

28. Using double angle formulae to prove identities We can use the double angle formulae to prove other identities involving multiple angles. For example:

29. Solve the following equations for the given intervals. Give answers correct to the nearest whole degree where appropriate. Where radians are required, exact answers should be given. Exercise 1. 2. 3.

30. Solution: ANS:1. or

31. Solution: ANS: or 2.

32. Solution: or 3. ANS:

33. Prove the following identities: 1. 2. 3. Exercise

34. 1. Prove Proof: = r.h.s. l.h.s. Solutions: (double angle formulae)

35. = r.h.s. 2. Prove Proof: l.h.s. Solutions: (double angle formulae)

36. = r.h.s. Solutions: 3. Prove (addition formula) Proof: (double angle formulae)

37. Activity: True or false worksheet

41. SUMMARY You need to remember the following results. The addition formulae are in your formula booklets and are written as Notice that the cos formulae have opposite signs on the 2 sides.

42. Using the Addition Formulae e.g. Prove the following: Proof: l.h.s. ( formulae (1) and (2) )

43. SUMMARY The double angle formulae are:

44. N.B. The formula link any angle with double the angle. For example, they can by used for and We use them to solve equations to prove other identities to integrate some functions

45. Proof: l.h.s. = = r.h.s. e.g. Prove that (addition formula) (double angle formulae)

46. SUMMARY The rearrangements of the double angle formulae for are They are important in integration so you should either memorise them or be able to obtain them very quickly.