# HOW TO EXPRESS PRODUCTS OF TRIG FUNCTIONS AS SUM OR DIFFERENCE

## About "How to Express Products of Trig Functions as Sum or Difference"

How to Express Products of Trig Functions as Sum or Difference :

Here we are going to see some example problems to show expressing the product of trig functions as sum or difference.

sin(A + B) = sin A cos B + cos A sin B  -------(1)

sin(A − B) = sin A cos B − cos A sin B  -------(2)

cos(A + B) = cos A cos B − sin A sin B ------(3)

cos(A − B) = cos A cos B + sin A sin B  -------(4)

(1) + (2)

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

(1) - (2)

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

(3) + (4)

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

(3) - (4)

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

## Expressing Products of Trig Functions as Sum or Difference

Question 1 :

Express each of the following as a sum or difference

(i) sin 35°cos 28°

Solution :

=  sin 35°cos 28°

Multiply and divide the given trigonometric ratio by 2.

=  (2/2) sin 35°cos 28°

=  (1/2) (2 sin 35°cos 28°)

It exactly matches the formula 2 sin A cos B

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

=  (1/2) [sin (35°+28°) + sin (35°-28°)]

=  (1/2) [sin 63°sin 7°]

(ii) sin 4x cos 2x

Solution :

=  sin 4x cos 2x

Multiply and divide the given trigonometric ratio by 2.

=  (2/2) sin 4x cos 2x

=  (1/2) (2 sin 4x cos 2x)

It exactly matches the formula 2 sin A cos B

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

=  (1/2) [sin (4x+2x) + sin (4x-2x)]

=  (1/2) [sin 6x sin 2x]

(iii) 2 sin 10θ cos 2θ

Solution :

=  2 sin 10θ cos 2θ

It exactly matches the formula 2 sin A cos B

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

=  (1/2) [sin (10θ+2θ) + sin (10θ+2θ)]

=  (1/2) [sin 12θ sin 8θ]

(iv) cos 5θ cos 2θ

Solution :

=  cos 5θ cos 2θ

Multiply and divide the given trigonometric ratio by 2.

=  (2/2) cos 5θ cos 2θ

=  (1/2) (2 cos 5θ cos 2θ)

It exactly matches the formula 2 cos A cos B

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

=  (1/2) [cos (5θ + ) + cos (5θ - )]

=  (1/2) [cos 7θ + cos 3θ]

(v) sin 5θ sin 4θ.

Solution :

=  sin 5θ sin 4θ

Multiply and divide the given trigonometric ratio by 2.

=  (-2/-2) sin 5θ sin 4θ

=  (-1/2) (-2 sin 5θ sin 4θ)

It exactly matches the formula -2 sin 5θ sin 4θ

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

=  (-1/2) [cos (5θ + 4θ) - cos (5θ - 4θ)]

=  (-1/2) [cos 9θ - cos θ]

=  (1/2)[cos θ - cos 9θ]

After having gone through the stuff given above, we hope that the students would have understood, "How to Express Products of Trig Functions as Sum or Difference"

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