# How do you find the exact value of the six trigonometric functions of the angle whose terminal side passes through (x, 4x)?

May 10, 2017

$\sin \theta = \frac{4}{\sqrt{17}}$, $\cos \theta = \frac{1}{\sqrt{17}}$, $\tan \theta = 4$,

$\cot \theta = \frac{1}{4}$, $\sec \theta = \sqrt{17}$, $\csc \theta = \frac{\sqrt{17}}{4}$

#### Explanation:

As the terminal side passes through $\left(x , 4 x\right)$, we have $y = 4 x$ and hence its distance from origin is $\sqrt{{x}^{2} + {\left(4 x\right)}^{2}} = \sqrt{{x}^{2} + 16 {x}^{2}} = x \sqrt{17}$

Now consider the diagram below for a typical $\theta$, whose six trigonometrical ratios are

$\sin \theta = \frac{y}{r}$, $\cos \theta = \frac{x}{r}$, $\tan \theta = \frac{y}{x}$ and

$\cot \theta = \frac{x}{y}$, $\sec \theta = \frac{r}{x}$, $\csc \theta = \frac{r}{y}$ As we have $y = 4 x$ and $r = x \sqrt{17}$,

the six trigonometric ratios are

$\sin \theta = \frac{4 x}{x \sqrt{17}} = \frac{4}{\sqrt{17}}$, $\cos \theta = \frac{x}{x \sqrt{17}} = \frac{1}{\sqrt{17}}$,

$\tan \theta = \frac{4 x}{x} = 4$, $\cot \theta = \frac{x}{4 x} = \frac{1}{4}$,

$\sec \theta = \frac{x \sqrt{17}}{x} = \sqrt{17}$, $\csc \theta = \frac{x \sqrt{17}}{4 x} = \frac{\sqrt{17}}{4}$