1. 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 What does sin 2A = ? Setting A = B

2. 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

3. 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.

4. SUMMARY The double angle formulae are:

5. 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

6. Activity: Trig double angle Card match

7. Using double angle formulae to solve equations We can use the double angle formulae to solve equations involving multiple angles. For example:

8. Solution: ANS: or

9. 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.

10. Solution: ANS: or 1.

11. Solution: or 2. ANS:

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

13. Prove the following identities: 1. 2. Exercise

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

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

16. Activity: True or false worksheet

19. SUMMARY The double angle formulae are:

20. 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

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

22. 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.