2. (a) To use the formulae sin (A B), cos (A B) and tan (A B). (b) To derive and use the double angle formulae (c) To derive and use the half angle formulae (d) Find the value of an angle without using table or calculator

3. Compound Angles Formulae + +

4. If we substitute B with A into the compound angle formulae, we have sin(A + A) cos(A + A) tan(A + A)

5. Double Angle Formulae sin 2A cos 2A tan 2A Thus, or

6. Half- Angle Formulae sin A cos A tan A or By substituting A with in the double angle formulae.

8. Example 1 If and, A and B are acute angles, find without using calculators, the value of a) sin (A+B) b) cos (A-B) c) tan (A+B)

9. Solution 3 4 5 A B 5 12 13 Given

10. a) sin (A+B)

11. b) cos (A-B)

12. c) tan (A+B)

13. Example 2 Simplify: a) sin 4x cos x – cos 4x sin x Solution a) sin 4x cos x – cos 4x sin x = sin (4x – x) = sin 3x

14. = tan (2x –x) = tan x

15. Example 3 Find the values of the followings without using calculator: a) cos 170 o cos 70 o – sin 170 o sin 70 o = cos ( 170 o + 70 o ) = cos 240 o = - cos 60 o 240 o 60 o -ve

16. b) = 3

17. Example 4 Find the exact value: a) sin 15 o = sin (60 0 - 45 0 ) = sin 60 0 cos 45 0 - cos 60 o sin 45 o

19. Example 5 Given that where A in the fourth quadrant, find without using calculator: b) cos 2A a) tan 2A

20. Solution Given A in the fourth quadrant. 15 17 8 A

21. a) tan 2A

22. b) cos 2A

23. Since A is in the 4 th quadrant

24. d) sin A

25. Example 6 Find the exact value of = tan 135 o = - tan 45 o = -1 135 o 45 o

26. Since 22.5 o is in the 1 st quadrant

27. REMEMBER?????!!!!!

28. Let A = 45 o sin 22.5 o

29. Example 7 Prove the identity Solution

30. Example 8 Prove the identity

31. Solution

32. Example 9 Show that

33. Solution

35. Exercise:

36. + +