Question

The modulus amplitude form of $1+itan\theta$ is:

• A. $\sec \theta cis\theta$
• B. $cis\theta$
• C. $\sec \theta cis(\frac{\pi }{2}-\theta )$
• D. $\sec \theta .cis(\frac{3\pi }{2}-\theta )$

Answer 1

The correct option is A $\sec \theta cis\theta$

$z=1+itan\theta$
$=1+i\frac{sin\theta }{cos\theta }$
$=\frac{cos\theta +isin\theta }{cos\theta }$
$=sec\theta (cos\theta +isin\theta )$
$=sec\theta \left(e^{i\theta }\right)$
Hence
$|z|=sec\theta$
And
$arg\left(z\right)=\theta$.