# EXPRESSING PRODUCTS OF TRIG FUNCTIONS AS PRODUCT

## About "Expressing Products of Trig Functions as Product"

Expressing Products of Trig Functions as Product :

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

sin C + sin D  =  2 sin [(C + D)/2] cos [(C - D)/2]

sin C - sin D  =  2 cos [(C + D)/2] sin [(C - D)/2]

cos C + cos D  =  2 cos [(C + D)/2] cos [(C - D)/2]

cos C - cos D  =  -2 sin [(C + D)/2] sin [(C - D)/2]

## Expressing Products of Trig Functions as product

Question 1 :

Express each of the following as a product

(i) sin 75° − sin 35°

Solution :

=  sin 75° − sin 35°

It exactly matches the formula

sin C - sin D  =  2 cos [(C + D)/2] sin [(C - D)/2]

=  2 cos [(75 + 35)/2] sin [(75 - 35)/2]

=  2 cos (110/2) sin (40/2)

=  2 cos 55° sin 20°

(ii) cos 65° + cos 15°

Solution :

=  cos 65° + cos 15°

It exactly matches the formula

cos C + cos D  =  2 cos [(C + D)/2] cos [(C - D)/2]

=  2 cos [(65 + 15)/2] cos [(65 - 15)/2]

=  2 cos (80/2) cos (50/2)

=  2 cos 40° cos 25°

(iii) sin 50° + sin 40°

Solution :

=  sin 50° + sin 40°

It exactly matches the formula

sin C + sin D  =  2 sin [(C + D)/2] cos [(C - D)/2]

=  2 sin [(50 + 40)/2] cos [(50 - 40)/2]

=  2 cos (90/2) cos (10/2)

=  2 cos 45° cos 5°

(iv) cos 35° − cos 75°

Solution :

=  cos 35° − cos 75°

It exactly matches the formula

cos C - cos D  =  -2 sin [(C + D)/2] sin [(C - D)/2]

=  -2 sin [(35 + 75)/2] cos [(35 - 75)/2]

=  -2 cos (110/2) cos (-40/2)

=  -2 cos 55° cos (-20°)

=  -2 cos 55° cos 20°

cos (-θ)  =  cos θ

Question 2 :

Show that sin 12° sin 48° sin 54°  =  1/8

Solution :

=  sin 12° sin 48° sin 54°

=   sin (90 - 36) sin 12° sin 48°

=   sin (90 - 36) [(-2/-2)(sin 12° sin 48°)]

=   sin (90 - 36) [(1/-2)(-2sin 12° sin 48°)]

=   sin (90 - 36) [(1/-2) (cos (12 + 48) - cos (12 - 48))]

=   cos 36 [(1/-2) (cos 60 - cos 36]

=  (-1/2) cos 36 [(1/2) - cos 36]

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

=  (-1/2) (√5 + 1)/4 [(2 - √5 - 1)/4]

=  (-1/2) (1 + √5)/4 [(1 - √5)/4]

=  (-1/2) (1 - 5)/16

=  (-1/2) (-1/4)

=  1/8

After having gone through the stuff given above, we hope that the students would have understood, "Expressing Products of Trig Functions as Product"

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