The point $(8, 15)$ is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

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Solution

Step-1: Find the value of $r$ :
The given point $(8, 15)$ .
Draw a right triangle in the first quarter of the rectangular coordinate plane with the base is $8$ and the height is $15$ .
Using the Pythagorean Theorem, compute the hypotenuse:
$\begin{array}{rcl} r & = & \sqrt{8^{2} + 15^{2}} \\ = & \sqrt{64 + 225} \\ = & \sqrt{289} \\ = & 17 \end{array}$
Step-2: Find the value of the trigonometric functions of $\sin (θ)$ , $\cos (θ)$ and $\tan (θ)$ :
Determine the fundamental trigonometric functions by applying the trigonometric ratios, as shown below:
Find the value of $\sin (θ)$ :
$\begin{array}{rcl} \sin (θ) & = & \frac{o p p}{h y p} \\ = & \frac{y}{r} \\ = & \frac{15}{17} \end{array}$
The value of $\sin (θ)$ is $\frac{15}{17}$ .
Find the value of $\cos (θ)$ :
$\begin{array}{rcl} \cos (θ) & = & \frac{A d j}{h y p} \\ = & \frac{x}{r} \\ = & \frac{8}{17} \end{array}$
The value of $\cos (θ)$ is $\frac{8}{17}$ .
Find the value of $\tan (θ)$ :
$\begin{array}{rcl} \tan (θ) & = & \frac{O p p}{A d j} \\ = & \frac{y}{x} \\ = & \frac{15}{8} \end{array}$
The value of $\tan (θ)$ is $\frac{15}{8}$
Step-3: Find the value of the trigonometric functions of $\csc (θ)$ , $s e c (θ)$ and $c o t (θ)$ :
Find the value of $\csc (θ)$ :
$\begin{array}{rcl} \csc (θ) & = & \frac{1}{\sin (θ)} \\ = & \frac{1}{\frac{15}{17}} \\ = & \frac{17}{15} \end{array}$
The value of $\csc (θ)$ is $\frac{17}{15}$ .
Find the value of $s e c (θ)$ :
$\begin{array}{rcl} s e c (θ) & = & \frac{1}{\cos (θ)} \\ = & \frac{1}{\frac{8}{17}} \\ = & \frac{17}{8} \end{array}$
The value of $s e c (θ)$ is $\frac{17}{8}$
Find the value of $c o t (θ)$ :
$\begin{array}{rcl} c o t (θ) & = & \frac{1}{\tan (θ)} \\ = & \frac{1}{\frac{15}{8}} \\ = & \frac{8}{15} \end{array}$
The value of $c o t (θ)$ is $\frac{8}{15}$ .
Hence, the exact values of the six trigonometric functions of the angle are $\frac{15}{17}, \frac{8}{17}, \frac{15}{8}, \frac{17}{15}, \frac{17}{8}, \frac{8}{15} .$

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The point (8,15) is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?

The point $(8, 15)$ is on the terminal side of an angle in standard position, how do you determine the exact values of the six trigonometric functions of the angle?