If-and-are-the-roots-of-the-equation-determinant-x-cos-sin-sin-x-cos-find-the-value-of-n-n-where-n-N-

Question Number 143150 by ZiYangLee last updated on 10/Jun/21

If α and β are the roots of the equation

| \begin{matrix} x - \cos θ & - \sin θ \\ \sin θ & x - \cos θ \end{matrix} |,

find the value of α^{n} + β^{n}, where n \in N .

Commented by mr W last updated on 10/Jun/21

| \begin{matrix} x - \cos θ & - \sin θ \\ \sin θ & x - \cos θ \end{matrix} | i s n o t a n e q u a t i o n .

{i t}^{'} s o n l y a v a l u e .

m a y b e y o u m e a n :

| \begin{matrix} x - \cos θ & - \sin θ \\ \sin θ & x - \cos θ \end{matrix} | = 0

Commented by Dwaipayan Shikari last updated on 10/Jun/21

{(x - c o s θ)}^{2} + {s i n}^{2} θ = 0

\Rightarrow x = c o s θ \pm i s i n θ = e^{\pm i θ}

α^{n} + β^{n} = e^{i n θ} + e^{- i n θ} = 2 c o s (n θ)