# Session Objectives

At the end of this chapter, candidates should be able to use the product formulae to solve trigonometric problems.

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# Product Formulae

## First Expression

Recall that from the addition formulae,

sin (A + B) = sinAcosB + cosAsinB

And

sin (A − B) = sinAcosB − cosAsinB

Adding the two expressions together will give us;

sin (A + B) = sinAcosB + cosAsinB

+

sin (A − B) = sinAcosB − cosAsinB

sin (A + B) + sin (A − B) = sinAcosB + cosAsinB + sinAcosB − cosAsinB

sin (A + B) + sin (A − B) = 2sinAcosB

## Second Expression

sin (A + B) = sinAcosB + cosAsinB

And

sin (A − B) = sinAcosB − cosAsinB

Subtracting the two expressions will give us;

sin (A + B) = sinAcosB + cosAsinB

−

sin (A − B) = sinAcosB − cosAsinB

sin (A + B) − sin (A − B) = sinAcosB + cosAsinB − (sinAcosB − cosAsinB)

sin (A + B) − sin (A − B) = sinAcosB + cosAsinB − sinAcosB + cosAsinB

sin (A + B) − sin (A − B) = 2cosAsinB

Let

X = 2sinAcosB,

Y = 2cosAsinB

P = sin (A + B)

Q = sin (A − B)

Hence,

X = P + Q and,

Y = P − Q

X + Y = P + Q + P − Q

X + Y = 2P

Make **P** the subject of the formula;

X + Y = 2Ptop↑

X = P + Q

Y = P − Q

X − Y = P + Q − (P − Q)

X − Y = P + Q − P + Q

X − Y = 2Q

Make

**Q**the subject of the formula;

sin (A + B) + sin (A − B) = 2sinAcosB

And

sinX + sin Y = ½(2sin P + Qcos P − Q)

Recall that from the second expression;

sin (A + B) − sin (A − B) = 2cosAsinB top↑

And

sinX − sin Y = ½(2cos P + Qsin P − Q)

Also,

cos (A + B) = cosAcosB − sinAsinB

And,

cos (A − B) = cosAcosB + sinAsinB

If we put X = P + Q, and Y = P − Q

then,

Hence,

cos X + cos Y = ½(2cos P + Qcos P − Q)

cos X − cos Y = ½(2sin P + Q sin P − Q)

All the above formulae that we just expressed are called **Product Formulae**

Now let look at 2 or more examples,

# Examples Of Product Of Two Trigonometric Ratio

## Example 1

Express sin 4x + sin 2x and sin 8x − sin 2x as product of two trigonometric ratios.

### Solution 1

To solve this question, you don't need any calculations just follow the expressions that will have analyse above and input the parameters.top↑

sin 4x + sin 2x = ½(2sin 4x + 2x cos 4x − 2x)

= 2sin 3x cos x

### Solution 2

Also use the same procedure

sin 8x − sin 2x = ½ (2cos8x + 2x sin 8x − 2x)

= 2cos 5x sin 3x

## Example 2

Express cos 6x + cos 4x and cos 4x − cos 2x as product of two trigonometric ratios.

### Solution 3

cos 6x + cos 4x = ½( 2cos 6x + 4xcos 6x − 4x)

= 2cos 5x cos x

### Solution 4

cos 4x − cos 2x = ½(−2sin 4x + 2xsin 4x − 2x)

−2sin 3x sin x

# Examples Of Sum Of Two Trigonometric Ratio

## Example 1

Express sin 5xcos 3x as a sum of two trigonometric ratio.top↑

### Solution

Recall,

sinX + sinY = ½(2sin X + Ycos X − Y)

Agree that

½[sinX + sinY] = ½(sin X + Ycos X − Y)

Put

And

then cross multiply,

X + Y = 5x × 2

X + Y = 10x....... eqn (1)

X − Y = 3x × 2

X − Y = 6x........ eqn (2)

Solving the two equation simultaneously

X + Y = 10x

X − Y = 6xtop↑

Making **X** the subject of the formula in equation 2

X − Y = 6x

X = 6x + Y

Put **X = 6x + Y** into equation 1

X + Y = 10x

(6x + Y) + Y = 10x

6x + 2Y = 10x

2Y = 10x − 6x

2Y = 4x

Y = 2x

Put **Y = 2x** into equation 2

X − Y = 6x

X − (2x) = 6x

X − 2x = 6x

X = 6x + 2x

X = 8x

So if X = 8x and Y = 2x then,

sin 5xcos 3x = ½[sin 8x + sin 2x]

## Example 2 Express cos 7xsin 5x as a sum of two trigonometric ratio.

### Solution

Similar to the first example,

cos 7xsin 5x = ½[sinX − sinY]top↑

Put

And

then cross multiply,

X + Y = 7x × 2

X + Y = 14x...... eqn (1)

X − Y = 5x × 2

X − Y = 10x ........ eqn (2)

Solving the two equations simultaneously,

X + Y = 14x

X − Y = 10x

Making X the subject of the formula in equation 1

X + Y = 14x

X = 14x − Y

Put **X = 14x − Y** into equation 2

X − Y = 10x

(14x − Y) − Y = 10x

14x − Y − Y = 10x

14x − 2 Y = 10x

−2Y = 10x − 14x

−2Y = − 4x

Y = 2xtop↑

Put **Y = 2x ** into equation 1

X + Y = 14x

X + (2x) = 14x

X + 2x = 14x

X = 14x − 2x

X = 12x

So if X = 12x and Y = 2x then,

cos 7xsin 5x = ½[sin 12x − sin2x]

## Example 3

Express cos 9xcos 3x as a sum of two trigonometric ratio.

### Solution

Same procedure,

cos 9xcos 3x = ½[cosX + cosY]

Put

And

then cross multiply,

X + Y = 9x × 2

X + Y = 14x....... eqn (1)

X − Y = 3x × 2

X − Y = 6x........ eqn (2)

Solving the two equations simultaneously

X + Y = 14x

X − Y = 6x

Make **X or Y** the subject of the formula in any of the equation. (Your choice)

Making **X** the subject of the formula in equation 2top↑

X − Y = 6x

X = 6x + Y

Put **X = 6x + Y** into equation 1

X + Y = 14x

(6x + Y) + Y = 14x

6x + Y + Y = 14x

6x + 2Y = 14x

2Y = 14x − 6x

2Y = 12x

Y = 6x

Put **Y = 6x** into equation 2

X − Y = 6x

X − (6x) = 6x

X − 6x = 6x

X = 6x + 6x

X = 12x

So if X = 12x and Y = 6x then,

cos 9x cos 3x = ½[cos 12x + cos 6x]

# Question Of The Day

Express sin 3xsinx as a sum of two trigonometric ratio? Submit your answer through the comment box.

# Jokes Of The Day

I can't stop laughing when I saw this picture, lol, it is true tho...

Is this true?top↑

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