Question

Find the values of the six trigonometric functions of an angle in standard position if the point with coordinates (84, 13) lies on its terminal side.

Answer 1

Answer:

$\sin \theta = 0.153$, $\cos \theta = 0.988$, $\tan \theta = 0.155$, $\cot \theta = 6.462$, $\sec \theta = 1.012$, $\csc \theta = 6.538$

Step-by-step explanation:

Let be the point $(x,y)$, the six trigonometric functions of the angle are represented by following formulas:

$\sin \theta = \frac{y}{\sqrt{x^{2}+y^{2}}}$ (1)

$\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}$ (2)

$\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{y}{x}$ (3)

$\cot \theta = \frac{1}{\tan \theta} = \frac{x}{y}$ (4)

$\sec \theta = \frac{1}{\cos \theta} = \frac{\sqrt{x^{2}+y^{2}}}{x}$ (5)

$\csc \theta = \frac{1}{\sin \theta} = \frac{\sqrt{x^{2}+y^{2}}}{y}$ (6)

If we know that $x = 84$ and $y = 13$, then the values of the six trigonometric functions is:

$\sin \theta = \frac{y}{\sqrt{x^{2}+y^{2}}}$

$\sin \theta = 0.153$

$\cos \theta = \frac{x}{\sqrt{x^{2}+y^{2}}}$

$\cos \theta = 0.988$

$\tan \theta = \frac{y}{x}$

$\tan \theta = 0.155$

$\cot \theta = \frac{x}{y}$

$\cot \theta = 6.462$

$\sec \theta = \frac{\sqrt{x^{2}+y^{2}}}{x}$

$\sec \theta = 1.012$

$\csc \theta = \frac{\sqrt{x^{2}+y^{2}}}{y}$

$\csc \theta = 6.538$