The expression $\frac{(\text{cis}\ 75^\circ)(\text{cis}\ 80^\circ−1)}{\text{cis}\ 8^\circ−1}$ can be written as $r\ \text{cis}\ \theta$.

Question

$\text{cis}(θ)=\cos θ+i\sin θ$. The expression $\dfrac{(\text{cis}\ 75^\circ)(\text{cis}\ 80^\circ−1)}{\text{cis}\ 8^\circ−1}$ can be written as $r\ \text{cis}\ \theta$ where $0 \le \theta < 360^\circ$. Find $\theta$ in degrees.

Answer 1

HINT:

cis$(2y)-1=\cos2y+i\sin2y-1=-2\sin^2y+2i\sin y\cos y$

$=2i\sin y(\cos y+i\sin y)=2i\sin y\cdot $cis$(y)$

Here $2y=8^\circ,80^\circ$

We may use : How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?