Question

Exactly find sin(x+y) if sinx= 1/3 where x ends in the

2ndquadrant and cosy= 1/5 where y ends in the first quadrant.

Answer 1

Answer:
`\sin(x+y)=(1-8√(3))/(15)`

Explanation:

Since we have given that

`\sin x=(1)/(3)\\\\so,\\\\\cos x=\sqrt{1-(1)/(9)}=\sqrt{(8)/(9)}=(2√(2))/(3)`

Since x ends in the 2 nd quadrant,

So,
`\cos x=(-2√(2))/(3)`

Similarly,

`\cos y=(1)/(5)\\\\So,\\\\\sin y=\sqrt{1-(1)/(25)}=\sqrt{(24)/(25)}=(2√(6))/(5)`

So, sin(x+y) is given by

`\sin x\cos y+\sin y\cos x\\\\\\=(1)/(3)* (1)/(5)+(2√(6))/(5)* (-)(2√(2))/(3)\\\\\\=(1)/(15)-(8√(3))/(15)\\\\\\=(1-8√(3))/(15)`

Hence,
`\sin(x+y)=(1-8√(3))/(15)`