Question

Given $sec\theta =\frac{13}{12}$, calculate all other trigonometric ratios

Answer 1

$\sec \theta =\frac{13}{12}\Rightarrow \cos \theta =\frac{1}{\sec \theta }=\frac{12}{13}$

$\sin \theta =\sqrt{1-{\cos }^2\theta }=\sqrt{1-{\left(\frac{12}{13}\right)}^2}=\sqrt{1-\frac{144}{169}}=\sqrt{\frac{25}{169}}=\pm \frac{5}{13}$

Note$\cos \theta$ is positive in first and fourth quadrant and $\sin \theta$ is positive in first quadrant and negative in third quadrant.

$\csc \theta =\frac{1}{\sin \theta }=\frac{1}{\frac{\pm 5}{13}}=\pm \frac{13}{5}$

$\tan \theta =\frac{\sin \theta }{\cos \theta }=\frac{\pm \frac{13}{5}}{\frac{12}{13}}=\frac{\pm 13}{12}$

$\cot \theta =\frac{1}{\tan \theta }=\pm \frac{12}{13}$