HOW TO EXPRESS 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 + 2θ) + cos (5θ - 2θ)]

  =  (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]

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