Question

If sine of the quantity x plus y end quantity equals one half times sine of x plus radical 3 over 2 times cosine of x comma what is the value of y?

Answer 1

The sine addition formula states that

$\sin (a+b)=\sin a\cos b+\cos a\sin b$

Thus, in our case,

$\sin (x+y)=\sin x\cos y+\cos x\sin y$

Then,

$\begin{gathered} \Rightarrow\sin x\cos y+\cos x\sin y=\frac{1}{2}\sin x+\frac{\sqrt[]{3}}{2}\cos x \\ \Rightarrow\cos y=\frac{1}{2},\sin y=\frac{\sqrt[]{3}}{2} \end{gathered}$

Solving for y,

$\Rightarrow y=\cos ^{-1}(\frac{1}{2})=\frac{\pi}{3}$

Thus, the answer is y=pi/3