8.3 Q-1 | Fundamentals Of Trigonometry | Mathematics 11

Exercise 8.3 — Q1: Product-to-Sum Identities

The four product-to-sum identities convert products of sines and cosines into sums or differences:

$2 sin A cos B = sin (A + B) + sin (A - B)$

$2 cos A sin B = sin (A + B) - sin (A - B)$

$2 cos A cos B = cos (A - B) + cos (A + B)$

$2 sin A sin B = cos (A - B) - cos (A + B)$

Example 1: Express $2 sin 75° cos 15°$ as a sum or difference.

Using $2 sin A cos B = sin (A + B) + sin (A - B)$ with $A = 75°$ , $B = 15°$ :

$2 sin 75° cos 15° = sin (90°) + sin (60°) = 1 + \frac{3}{2}$

Example 2: Express $6 cos 5 x sin 10 x$ as a sum or difference.

Factor out 3: $6 cos 5 x sin 10 x = 3 \cdot 2 cos 5 x sin 10 x$

Using $2 cos A sin B = sin (A + B) - sin (A - B)$ with $A = 5 x$ , $B = 10 x$ :

$= 3 [sin (15 x) - sin (- 5 x)] = 3 [sin (15 x) + sin (5 x)]$

(since $sin (- 5 x) = - sin (5 x)$ )

Example 3: Express $2 cos 3 θ cos θ$ as a sum or difference.

Using $2 cos A cos B = cos (A - B) + cos (A + B)$ with $A = 3 θ$ , $B = θ$ :

$2 cos 3 θ cos θ = cos (2 θ) + cos (4 θ)$

Example 4: Express $2 sin 4 x sin 2 x$ as a sum or difference.

Using $2 sin A sin B = cos (A - B) - cos (A + B)$ with $A = 4 x$ , $B = 2 x$ :

$2 sin 4 x sin 2 x = cos (2 x) - cos (6 x)$