If the real part of z=3+2i1-3i, [0,2], is zero, then ^23+^2 is equal to Answer: 1

Question

If the real part of $z=\dfrac{3+2i\cos	heta}{1-3i\cos	heta}$, $	heta\in\left[0,\dfrac\pi2ight]$, is zero, then $\sin^23	heta+\cos^2	heta$ is equal to

Answer: 1

Accepted Answer

Rationalizing gives $\operatorname{Re}(z)=\dfrac{3-6\cos^2\theta}{1+9\cos^2\theta}=0 \implies \cos^2\theta=\dfrac12 \implies \theta=\dfrac\pi4.$

$$\sin^2\!\left(\frac{3\pi}4\right)+\cos^2\!\left(\frac\pi4\right)=\frac12+\frac12=1.$$

Solutions · 1