Mathematics : Product Formulae

mathematics product formulae

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↑

P = X + Y
       2

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;

Q = X − Y
      2
Recall that from the first expression;
sin (A + B) + sin (A − B) = 2sinAcosB

And

Q = X − Y
       2
P = X + Y
       2
then,
sinX + sin Y = ½(2sin P + Qcos P − Q)

Recall that from the second expression;
sin (A + B) − sin (A − B) = 2cosAsinB top↑

And

Q = X − Y
       2
P = X + Y
       2
then,
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,

Q = X − Y
        2
P
= X + Y
  2
top↑
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

5x = X + Y
         2

And

3x = X − Y
        2

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

7x = X + Y
        2

And

5x = X − Y
        2

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

9x = X + Y
        2

And

3x = X − Y
         2

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…
Jokes Of the day
Is this true?top↑

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