Question

If $\frac{3+i\,sin\theta}{4-i\,cos\theta}, \,\theta\,\epsilon\,\left[0,\,2\pi\right],$ is a real number, then an argument of $sin\theta + icos\theta$ is :

• A. $\pi-tan^{-1}\left(\frac{4}{3}\right)$
• B. $-tan^{-1}\left(\frac{3}{4}\right)$
• C. $\pi-tan^{-1}\left(\frac{3}{4}\right)$
• D. $tan^{-1}\left(\frac{4}{3}\right)$

Answer 1

Answer: (A)

$\frac{3+i\,sin\,\theta}{4-i\,cos\,\theta}$ is a real number
$\Rightarrow 3\,cos\,\theta +4\,sin\,\theta =0$
$\Rightarrow tan\,\theta = \frac{-3}{4}$
argument of $sin\,\theta +icos\,\theta =\pi-tan^{-1} \frac{4}{3}$