# If sin A = -513,π<A<3π2 and cos B = 35,3π2<B<2π find sin (A + B) - Mathematics and Statistics

Sum

If sin A = (-5)/13, pi < "A" < (3pi)/2 and cos B = 3/5, (3pi)/2 < "B" < 2pi find sin (A + B)

#### Solution

Given, sin A = (-5)/13

We know that,

cos2A = 1 – sin2A = 1 - (-5/13)^2

= 1 - 25/169

= 144/169

∴ cos A = ±12/13

Since, pi < "A" < (3pi)/2

∴ ‘A’ lies in the 3rd quadrant

∴ cos A < 0

∴ cos A = (-12)/13

Also, cos B = 3/5

∴ sin2B = 1 – cos2B = 1 - (3/5)^2

= 1 - 9/25

= 16/25

∴ sin B = ±4/5

Since, (3pi)/2 < "B" < 2pi

∴ ‘B’ lies in the 4th quadrant.

∴ sin B < 0

∴ sin B = (-4)/5

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

= (-5/13) (3/5) + (-12/13)(-4/5)

= -15/65 + 48/65

= 33/65

Concept: Trigonometric Functions of Sum and Difference of Angles
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