Question

tan(x)=2/3 and sec(x)=-sqrt13/3

Answer 1

Question #25414

$$\tan x = \frac {2}{3} \Rightarrow x = \pi n + \tan^ {- 1} \left(\frac {2}{3}\right), \qquad n \in \mathbb {Z}$$ $$\sec x = - \frac {\sqrt {13}}{3} \sim \frac {1}{\cos x} = - \frac {\sqrt {13}}{3} \sim \cos x = - \frac {3}{\sqrt {13}} \Rightarrow x = 2 \pi n \pm \cos^ {- 1} \left(- \frac {3}{\sqrt {13}}\right), \qquad n \in \mathbb {Z}$$ $$1 + \tan^ {2} x = \frac {1}{\cos^ {2} x}; 1 + \tan^ {2} x = \sec^ {2} x$$

Substitute with original data:

$$1 + \left(\frac {2}{3}\right) ^ {2} = \left(- \frac {\sqrt {13}}{3}\right) ^ {2}; 1 + \frac {4}{9} = \frac {13}{9}; \frac {13}{9} = \frac {13}{9}$$

We get a true equality for all $x$ . But we need to check signs (because $\sec(x)$ was with “-”). For answer of system equations we can take both answers of original equations. Also look for preservation period and sign.

$$\left\{ \begin{array}{c} \tan x = \frac {2}{3} \\ \sec x = - \frac {\sqrt {13}}{3} \end{array} \right. \left\{ \begin{array}{c} x = \pi n + \tan^ {- 1} \left(\frac {2}{3}\right) \\ x = 2 \pi n \pm \cos^ {- 1} \left(- \frac {3}{\sqrt {13}}\right) \end{array} \right. \Rightarrow x = 2 \pi n - \pi + \tan^ {- 1} \left(\frac {2}{3}\right), \qquad n \in \mathbb {Z}$$