If $A, B, C$ are angles of a triangle such that $x = cis A, y = cis B, z = cis C$ , then find the value of $x y z$ .

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Solution

Using Euler rormula $cis (θ) = e^{i θ}$

So $x = e^{i A}, y = e^{i B}$ and $z = e^{i C}$

Hence $x y z = e^{i A} \cdot e^{i B} \cdot e^{i C} = e^{i (A + B + C)} = e^{i (π)}$ (Since $A, B, C$ are angles of a triangle so, $A + B + C = π$ )

$= cos π + i sin π = - 1 + 0 = - 1$

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