Animated Solution for Mathematics - Complex Numbers: If $\frac{3+i\sin \theta }{4-i\cos \theta }$, $\theta \in [0,2\pi ]$, is a real number, then an argument of $\sin \theta +i\cos \theta$ is:

We are given the complex number $z=\frac{3+i\sin \theta }{4-i\cos \theta }$. We are informed that $z$ is a purely real number.

Our objective is to determine the argument of the complex number $w=\sin \theta +i\cos \theta$.

To ensure $z$ is purely real, its imaginary part must be zero. We begin by rationalizing the denominator using the conjugate $4+i\cos \theta$:

The denominator simplifies to a real value:

Expanding the numerator, we obtain:

Since $i^2=-1$, the expression becomes:

For $z$ to be purely real, the imaginary part must vanish:

This yields the condition:

We now examine $w=\sin \theta +i\cos \theta$. Note that the real part is $\sin \theta$ and the imaginary part is $\cos \theta$.

The argument of $w$ is given by $\text{arg}(w)={\tan }^{-1}\left(\frac{\cos \theta }{\sin \theta }\right)={\tan }^{-1}(\cot \theta )$.

Given $\tan \theta =-\frac{3}{4}$, we have $\cot \theta =-\frac{4}{3}$.

Since $\tan \theta <0$, $\theta$ lies in the second or fourth quadrant. In the second quadrant, $\sin \theta >0$ and $\cos \theta <0$, which corresponds to the point $(\sin \theta ,\cos \theta )$ being in the fourth quadrant of the Argand plane.

The argument is:

Using the property ${\tan }^{-1}(-x)=-{\tan }^{-1}(x)$, the final result is: