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
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]
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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