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Given: $\sin{A}= \frac{4}{5}$ and $\cos{B} = \frac{-5}{13}$; evaluate the following expression. (A: Quadrant I, B : Quadrant II.)?

1. $\sin{( A+ B )}$
2. $\cos{( A+ B )}$
3. $\sin{( A+ A )}$
4. $\cos{( A+ A )}$
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## 1 Answer

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Given sin A= 4/5 = O/H , using Pythagoras ,

b = √(5^2 - 4^2) = 3

cos B = -5/13 = A/H

b = √(13^2 - 5^2) = 12

1. sin(A+B) , using compound angles

= sinAcosB + cosAsinB

= 4/5 x -5/13 + 3/5 x 12/13= 0.246... = 0.25

2. cos(A+B) = cosAcosB - sinAsinB

= 3/5 x -5/13 - 4/5 x 12/13 = -0.969... = -0.97

3. sin(A+A), using double angle,

= sin2A = 2sinAcosA

= 2 x 4/5 x 3/5

= 0.96

4. cos(A+A) = cos2A = 1 - 2sin^A

= 1 - 2 x (4/5)^2

= -0.28
by Diamond (42,470 points)

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