If $x + iy = \frac{3}{2 + \cos heta + i\sin heta}$, then what is the value of $x$?

Question

If $x + iy = \frac{3}{2 + \cos	heta + i\sin	heta}$, then what is the value of $x$?

Accepted Answer

Rationalize the denominator: $x + iy = \frac{3(2 + \cos	heta - i\sin	heta)}{(2 + \cos	heta)^2 + \sin^2	heta} = \frac{3(2 + \cos	heta - i\sin	heta)}{4 + 4\cos	heta + \cos^2	heta + \sin^2	heta} = \frac{3(2 + \cos	heta - i\sin	heta)}{5 + 4\cos	heta}$. Equating the real part gives $x = \frac{3(2 + \cos	heta)}{5 + 4\cos	heta}$.

Complex Numbers and Quadratic Equations

❓ Sibling Questions in this Chapter