Question

How do you find the exact values of the six trigonometric function of $\theta$ if the terminal side of $\theta$ in the standard position contains the point $\left(5,-8\right)$?

Answer 1

Finding the value of six trigonometric functions of $\theta$:

Step 1: Finding the position of $\theta$.

Given that the terminal side of $\theta$ in the standard position contains the point $\left(5,-8\right)$.

Now, the point $\left(5,-8\right)$ lies in the fourth quadrant. So, the angle $\theta$ lies in the fourth quadrant.

Step 2: Finding the value of $\tan \theta$.

We have: $x=5,y=-8$ and we know that

$$\begin{gathered} \tan \theta =\frac{y}{x} \\ =-\frac{8}{5} \end{gathered}$$

Step-3: Finding the value of $sec\theta$.

We know that $se{c}^2\theta =1+{\tan }^2\theta$.

Thus, we get:

$$\begin{gathered} se{c}^2\theta =1+{\tan }^2\theta \\ =1+{\left(-\frac{8}{5}\right)}^2 \\ =1+\frac{64}{25} \\ =\frac{89}{25} \\ \Rightarrow sec\theta =\sqrt{\frac{89}{25}} \\ \therefore sec\theta =\frac{\sqrt{89}}{5}\left[\text{as}\theta \text{lies in fourth quadrant,}sec\theta >0\right] \end{gathered}$$

Step 4: Finding the value of $\cos \theta$.

We know that $\cos \theta =\frac{1}{sec\theta }$.

Therefore, we get:

$$\begin{gathered} \cos \theta =\frac{1}{sec\theta } \\ =\frac{1}{{\displaystyle \frac{\sqrt{89}}{5}}} \\ =\frac{5}{\sqrt{89}} \end{gathered}$$

Step 5: Finding the value of $\sin \theta$.

We know that ${\sin }^2\theta +{\cos }^2\theta =1$.

Thus, we get:

$$\begin{gathered} {\sin }^2\theta =1-{\cos }^2\theta \\ =1-{\left(\frac{5}{\sqrt{89}}\right)}^2 \\ =1-\frac{25}{89} \\ =\frac{64}{89} \\ \Rightarrow \sin \theta =-\sqrt{\frac{64}{89}}\left[\text{as}\theta \text{lies in the fourth quadrant,}\sin \theta <0\right] \\ \therefore \sin \theta =-\frac{8}{\sqrt{89}} \end{gathered}$$

Step 6: Finding the value of $\cos ec\theta$.

We know that $\cos ec\theta =\frac{1}{\sin \theta }$.

So, we get:

$$\begin{gathered} \cos ec\theta =\frac{1}{\sin \theta } \\ =\frac{1}{-{\displaystyle \frac{8}{\sqrt{89}}}} \\ =-\frac{\sqrt{89}}{8} \end{gathered}$$

Step 7: Finding the value of $\mathrm{co}t\theta$.

We know that $\mathrm{co}t\theta =\frac{1}{\tan \theta }$.

Thus,

$$\begin{gathered} \mathrm{co}t\theta =\frac{1}{\tan \theta } \\ =\frac{1}{-{\displaystyle \frac{8}{5}}} \\ =-\frac{5}{8} \end{gathered}$$

Therefore, the exact values of the six trigonometric functions of $\theta$ if the terminal side of $\theta$ in the standard position contains the point $\left(5,-8\right)$ are : $\sin \theta =-\frac{8}{\sqrt{89}},\cos \theta =\frac{5}{\sqrt{89}},\tan \theta =-\frac{8}{5},\cos ec\theta =-\frac{\sqrt{89}}{8},sec\theta =\frac{\sqrt{89}}{5},cot\theta =-\frac{5}{8}$.