Question

How do you find all six trigonometric functions of $\mathrm{\theta }$if the given point $(5,0)$ is on the terminal side of $\mathrm{\theta }$?

Answer 1

Step-1: Find the distance between the origin and the given point:

Given terminal point is $(5,0)$.

Let the distance between the origin and the terminal point be $r$

If the point in the angle’s terminal side is $\mathrm{P}=\left(\mathrm{x},\mathrm{y}\right)$ then the trigonometric functions can be calculated as:

$\sin \left(\theta \right)=\frac{\mathrm{y}}{\mathrm{r}}$, $\cos \left(\theta \right)=\frac{\mathrm{x}}{\mathrm{r}}$, $\tan \left(\theta \right)=\frac{\mathrm{y}}{\mathrm{x}}$, $\cot \left(\theta \right)=\frac{\mathrm{x}}{\mathrm{y}}$, $\sec \left(\theta \right)=\frac{\mathrm{r}}{\mathrm{x}}$, $\mathrm{cosec}\left(\theta \right)=\frac{\mathrm{r}}{\mathrm{y}}$, where $\mathrm{r}=\sqrt{{\mathrm{x}}^2+{\mathrm{y}}^2}$

Here, $\mathrm{x}=5,\mathrm{y}=0$

$\Rightarrow$$\mathrm{r}=\sqrt{{\left(5\right)}^2+{\left(0\right)}^2}$

$\Rightarrow$$\mathrm{r}=\sqrt{25}$

$\Rightarrow$$\mathrm{r}=5$

Step-2: Find the values of the six trigonometric functions:

Now substitute the values of $\mathrm{x},\mathrm{y}$ and $\mathrm{r}$ to find the value of all the trigonometric functions.

So, $\sin \left(\theta \right)=\frac{0}{5}$ $\left[\because \sin \left(\theta \right)=\frac{\mathrm{y}}{\mathrm{r}}\right]$

$\Rightarrow$ $\sin \left(\theta \right)=0$

$\cos \left(\theta \right)=\frac{5}{5}$ $\left[\because \cos \left(\theta \right)=\frac{\mathrm{x}}{\mathrm{r}}\right]$

$\Rightarrow$ $\cos \left(\theta \right)=1$

$\tan \left(\theta \right)=\frac{0}{5}$ $\left[\because \tan \left(\theta \right)=\frac{\mathrm{y}}{\mathrm{x}}\right]$

$\Rightarrow$ $\tan \left(\theta \right)=0$

$\cot \left(\theta \right)=\frac{5}{0}$ $\left[\because \cot \left(\theta \right)=\frac{\mathrm{x}}{\mathrm{y}}\right]$

$\Rightarrow$ $\cot \left(\theta \right)=\infty$ or undefined

$\sec \left(\theta \right)=\frac{5}{5}$ $\left[\because \sec \left(\theta \right)=\frac{\mathrm{r}}{\mathrm{x}}\right]$

$\Rightarrow$ $\sec \left(\theta \right)=1$

$\mathrm{cosec}\left(\theta \right)=\frac{5}{0}$ $\left[\because \mathrm{cosec}\left(\theta \right)=\frac{\mathrm{r}}{\mathrm{y}}\right]$

$\Rightarrow$ $\mathrm{cosec}\left(\theta \right)=\infty$or undefined

Hence, all the six trigonometric functions are calculated above.