Page 28 - DMTH201_Basic Mathematics-1

Unit 2: Trigonometric Functions-II

Notes
Example: Express the following products as a sum or difference
(i) 2sin 3 cos 2
(ii) cos 6 cos
Solution:
(i) (2sin 3 cos 2 = sin (3 + 2 ) + sin (3 2 )
= sin 5 + sin

(ii) cos 6 cos = 1/2 (2cos 6 cos )
= 1/2 [cos (6 + ) + cos (6 )]
= 1/2(cos 7 + cos 5 )

2.1.3 Trigonometric Functions of Multiples of Angles

(a) To express sin 2A in terms of sin A, cos A and tan A.

We know that
sin (A + B) = sin A cos B + cos A sin B
By putting B = A, we get
sin 2A = sin A cos A + cos A sin A
= 2sin A cos A

sin 2A can also be written as
(Q 1 = cos A + sin A)
2
2

Dividing numerator and denominator by cos A, we get
2

(b) To express cos 2A in terms of sin A, cos A and tan A.
We know that

cos (A + B) = cos A cos B – sin A sinB
Putting B = A, we have
cos 2A = cos A cos A – sin A sin A
or cos 2A = cos A – sin A
2
2
2
Also cos 2A = cos A – (1 – cos A)
2
2
2
= cos A – 1 + cos A
2
i.e, cos 2A = 2cos A – 1

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