8.3 Q-4 | Fundamentals Of Trigonometry | Mathematics 11

Exercise 8.3 — Question 4: Product-to-Sum Identities

This exercise applies the product-to-sum identities to express products of trigonometric functions as sums or differences.

Product-to-Sum Identities

The four fundamental product-to-sum identities are derived from the sum and difference formulas:

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

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

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

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

Worked Examples

Example 1: Express $6 cos 5 x sin 10 x$ as a sum.

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

$6 cos 5 x sin 10 x = 3 \cdot 2 cos 5 x sin 10 x$ $= 3 [sin (5 x + 10 x) - sin (5 x - 10 x)]$ $= 3 [sin 15 x - sin (- 5 x)]$ $= 3 [sin 15 x + sin 5 x]$

(since $sin (- θ) = - sin θ$ )

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

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

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

Example 3: Express $4 sin 5 x sin 3 x$ as a difference.

Using $2 sin A sin B = cos (A - B) - cos (A + B)$ with $A = 5 x$ , $B = 3 x$ :

$4 sin 5 x sin 3 x = 2 \cdot 2 sin 5 x sin 3 x = 2 [cos (5 x - 3 x) - cos (5 x + 3 x)] = 2 [cos 2 x - cos 8 x]$

Exercise 8.3 — Question 4: Product-to-Sum Identities

This exercise applies the product-to-sum identities to express products of trigonometric functions as sums or differences.

Product-to-Sum Identities

The four fundamental product-to-sum identities are derived from the sum and difference formulas:

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

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

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

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

Worked Examples

Example 1: Express $6 cos 5 x sin 10 x$ as a sum.

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

$6 cos 5 x sin 10 x = 3 \cdot 2 cos 5 x sin 10 x$ $= 3 [sin (5 x + 10 x) - sin (5 x - 10 x)]$ $= 3 [sin 15 x - sin (- 5 x)]$ $= 3 [sin 15 x + sin 5 x]$

(since $sin (- θ) = - sin θ$ )

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

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

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

Example 3: Express $4 sin 5 x sin 3 x$ as a difference.

Using $2 sin A sin B = cos (A - B) - cos (A + B)$ with $A = 5 x$ , $B = 3 x$ :

$4 sin 5 x sin 3 x = 2 \cdot 2 sin 5 x sin 3 x = 2 [cos (5 x - 3 x) - cos (5 x + 3 x)] = 2 [cos 2 x - cos 8 x]$