Question

If $A,B,C$ are angles of a triangle such that $x=\text{cis}A,y=\text{cis}B,z=\text{cis}C$, then find the value of $xyz=$

• A. $-1$
• B. $1$
• C. $-i$
• D. $i$

Answer 1

Answer: (A)

$-1$

Using Euler rormula $\text{cis}\left(\theta \right)=e^{i\theta }$

So $x=e^{iA},y=e^{iB}$ and $z=e^{iC}$

Hence $xyz=e^{iA}\cdot e^{iB}\cdot e^{iC}=e^{i(A+B+C)}=e^{i\left(\pi \right)}$

$=\cos \pi +i\sin \pi =-1+0=-1$

Since $A,B,C$ are angles of a triangle so, $A+B+C=\pi$