Question

Given sinx=1/3, cosy=3/4, and both x and y are in quadrant iv, find cos(x-y) and sin(x+y)

Answer 1

Answer:
`(6√(2)+√(7))/(12),\ (-3-2√(14))/(12)`

Explanation:

Given

x and y is in the fourth quadrant

`\sin x=-(1)/(3)\\\\\cos y=(3)/(4)`

we know,

`\cos (x-y)=\cos x\cos y+\sin x\sin y\\\sin (x+y)=\sin x\cos y+\cos x\sin y`

using
`\sin ^2\theta +\cos^2 \theta =1`

`\Rightarrow \cos x=(2√(2))/(3)\\\\\Rightarrow \sin y=-(√(7))/(4)`

`\Rightarrow \cos(x-y)=((2√(2))/(3))* ((3)/(4))+(-(1)/(3))* (-(√(7))/(4))=(6√(2)+√(7))/(12)\\\\\\\Rightarrow \sin (x+y)=(-(1)/(3))* (3)/(4)+(2√(2))/(3)* ((-√(7))/(4))=(-3-2√(14))/(12)`