1. Subtopic 2.3: Compound Angle Formulae
Lecture 4 of 12
Topic : Trigonometry
Subtopic 2.3: Compound Angle Formulae

2. Learning outcomes:
(a) express trigonometry products as sum
express trigonometry sums as products
(to derive and use factor formulae)

3. FACTOR FORMULAE sin(A+B) + sin(A-B) = 2sinAcosB …..(1)
sin(A+B) - sin(A-B) = 2cosAsinB …..(2)
cos(A+B) + cos(A-B) = 2cosAcosB …..(3)
cos(A+B) - cos(A-B) = -2sinAsinB …..(4)

4. To derive formulae 1 to 4, we use the compound angle formulae
sin(A+B) = sinAcosB + cosAsinB …..(a)
sin(A-B) = sinAcosB – cosAsinB …..(b)
(a) + (b),
sin(A+B) +sin(A-B) = 2sinAcosB
(a) - (b) ,
sin(A+B) – sin(A-B) = 2cosAsinB

5. Similarly for cos(A+B) and cos(A-B), we get
cos(A+B) + cos(A-B) = 2cosAcosB
cos(A+B) - cos(A-B) = -2sinAsinB
By substituting
A+B = M ……. (1)
and
A –B = N …… (2)
(1) + (2)
(1) – (2)

6. cos(A+B) + cos(A-B) = 2cosAcosB
By substituting
A+B = M
A –B = N
cos(A+B) + cos(A-B) = 2cosAcosB
cos M + cos N = 2cos cos

7. Similarly, we will obtain the NEW form of the factor formulae

8. Example 1
Find the following values without using calculator

9. Solution

12. Example 2
Express each sum or difference as a
product of sine or cosine.

13. solution

14. (b)

16. Example 3
Express each of the following products as a sum of sine or cosine.

17. Solution sin(A+B) + sin(A-B) = 2sinAcosB

19. Example 4
Show that
Solution

20. Example 5
Prove the following identities;
Solution

22. Example 6 Prove that
Solution
RHS :
Therefore

23. Alternative Method :
LHS :

24. Alternative Method :
RHS :
Therefore

25. Conclusions sin(A+B) + sin(A-B) = 2sinAcosB
sin(A+B) - sin(A-B) = 2cosAsinB
cos(A+B) + cos(A-B) = 2cosAcosB
cos(A+B) - cos(A-B) = -2sinAsinB