Some applications of trigonometric functions demand that a sum or difference of trigonometric functions be written as a product of trigonometric functions.
The identities given below will be useful to write the sum and difference of trigonometric 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]
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
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