Subtopic 2.3: Compound Angle Formulae Lecture 4 of 12 Topic : 2.0 Trigonometry Subtopic 2.3: Compound Angle Formulae

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

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)

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

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)

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

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

Example 1 Find the following values without using calculator

Solution

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

solution

(b)

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

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

Example 4 Show that Solution

Example 5 Prove the following identities; Solution

Example 6 Prove that Solution RHS : Therefore

Alternative Method : LHS :

Alternative Method : RHS : Therefore

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