(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

Compound Angles Formulae + +

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

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

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

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)

Solution A B Given

a) sin (A+B)

b) cos (A-B)

c) tan (A+B)

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

= tan (2x –x) = tan x

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

b) = 3

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

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

Solution Given A in the fourth quadrant A

a) tan 2A

b) cos 2A

Since A is in the 4 th quadrant

d) sin A

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

Since 22.5 o is in the 1 st quadrant

REMEMBER?????!!!!!

Let A = 45 o sin 22.5 o

Example 7 Prove the identity Solution

Example 8 Prove the identity

Solution

Example 9 Show that

Solution

Exercise:

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