8.3 Q-3 | Fundamentals Of Trigonometry | Mathematics 11

Exercise 8.3 — Question 3

This exercise applies product-to-sum identities to express products of sines and cosines as sums or differences.

Product-to-Sum Identities

The four fundamental product-to-sum formulas are:

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

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

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

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

Worked Examples

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

Solution:

Identify the form: $cos A \cdot sin B$ → use $2 cos A sin B = sin (A + B) - sin (A - B)$

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

$= 3 [sin (5 x + 10 x) - sin (5 x - 10 x)]$

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

$= 3 [sin 15 x + sin 5 x]$

(since $sin (- θ) = - sin θ$ )

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

Solution:

Use $2 sin A cos B = sin (A + B) + sin (A - B)$ 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 $2 cos 4 x cos 2 x$ as a sum.

Solution:

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

$2 cos 4 x cos 2 x = cos (4 x - 2 x) + cos (4 x + 2 x)$

$= cos 2 x + cos 6 x$

Example 4: Express $2 sin 5 α sin 3 α$ as a difference.

Solution:

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

$2 sin 5 α sin 3 α = cos (5 α - 3 α) - cos (5 α + 3 α)$

$= cos 2 α - cos 8 α$

Exercise 8.3 — Question 3

This exercise applies product-to-sum identities to express products of sines and cosines as sums or differences.

Product-to-Sum Identities

The four fundamental product-to-sum formulas are:

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

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

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

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

Worked Examples

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

Solution:

Identify the form: $cos A \cdot sin B$ → use $2 cos A sin B = sin (A + B) - sin (A - B)$

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

$= 3 [sin (5 x + 10 x) - sin (5 x - 10 x)]$

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

$= 3 [sin 15 x + sin 5 x]$

(since $sin (- θ) = - sin θ$ )

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

Solution:

Use $2 sin A cos B = sin (A + B) + sin (A - B)$ 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 $2 cos 4 x cos 2 x$ as a sum.

Solution:

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

$2 cos 4 x cos 2 x = cos (4 x - 2 x) + cos (4 x + 2 x)$

$= cos 2 x + cos 6 x$

Example 4: Express $2 sin 5 α sin 3 α$ as a difference.

Solution:

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

$2 sin 5 α sin 3 α = cos (5 α - 3 α) - cos (5 α + 3 α)$

$= cos 2 α - cos 8 α$