if tanx=1/5, cos2x=?
You asked:
Given \(\tan\left( x \right) = \frac{1}{5}\) evaluate the expression \(\cos\left( 2 x \right)\).
MathBot Answer:
The principal value in \(- \frac{\pi}{2} < x < \frac{\pi}{2}\) is \(x \approx 0.19739556\)
The real solutions are: \[\begin{aligned}x &= \left\{2 \pi n + \arctan{\left(\frac{1}{5} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 0.19739556\; \middle|\; n \in \mathbb{Z}\right\}\\x &= \left\{\pi + 2 \pi n + \arctan{\left(\frac{1}{5} \right)}\; \middle|\; n \in \mathbb{Z}\right\} \approx \left\{6.2831853 n + 3.3389882\; \middle|\; n \in \mathbb{Z}\right\}\end{aligned}\]
By substituting the principal value into the expression, we find that: $$\begin{aligned}\cos{\left(2 x \right)} \approx 0.92307692\end{aligned}$$