Question

Find the values of the trigonometric functions of $t$from the given information.

$\sec t=3$, terminal point of $t$is in Quadrant IV

Short answer

The values of the trigonometric functions are $\mathbf{sin}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{2}\sqrt{\mathbf{2}}}{\mathbf{3}}\mathbf{,}\mathbf{cos}\mathit{t}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}\mathbf{,}\mathbf{tan}\mathit{t}\mathbf{=}\mathbf{-}\mathbf{2}\sqrt{\mathbf{2}}$,$\mathbf{csc}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{3}}{\mathbf{2}\sqrt{\mathbf{2}}}$, and $\mathbf{cot}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{2}\sqrt{\mathbf{2}}}$.

Step-by-step solution

Step 1. State the Trigonometric identities.

The reciprocal identities are $\csc t=\frac{1}{\sin t}$, $\sec t=\frac{1}{\cos t}$, $\cot t=\frac{1}{\tan t}$, $\tan t=\frac{\sin t}{\cos t}$and $\cot =\frac{\cos t}{\sin t}$.

The Pythagorean identities are:

1. ${\sin }^{2}t+{\cos }^{2}t=1$

2. ${\tan }^{2}t+1={\sec }^{2}t$

3.${\cot }^{2}t+1={\csc }^{2}t$

Step 2. Formula used. 

To find $\cos t$, substitute 3 for $\sec t$into the reciprocal identity $\cos t=\frac{1}{\sec t}$.

$$\begin{gathered} \cos t=\frac{1}{\sec t} \\ =\frac{1}{3} \end{gathered}$$

Now use Pythagorean Identity ${\sin }^{2}t+{\cos }^{2}t=1$, substitute $\frac{1}{3}$for $\cos t$and simplify for $\cos t$.

$$\begin{gathered} {\sin }^{2}t+{\cos }^{2}t=1 \\ {\sin }^{2}t+{\left(\frac{1}{3}\right)}^2=1 \end{gathered}$$

$$\begin{gathered} {\sin }^{2}t+\frac{1}{9}-\frac{1}{9}=1-\frac{1}{9} \\ {\sin }^{2}t=\frac{8}{9} \end{gathered}$$

Now, $t$is given to be in quadrant IV, and quadrant IV, $\sin t<0$, therefore, $\sin t=-\frac{2\sqrt{2}}{3}$.

Step 3. Find other trigonometric functions.

Now substitute $-\frac{2\sqrt{2}}{3}$for $\sin t$and $\frac{1}{3}$for $\cos t$into reciprocal identity $\tan t=\frac{\sin t}{\cos t}$and simplify for $\tan t$.

$$\begin{gathered} \tan t=\frac{\sin t}{\cos t} \\ =\frac{\left(-\frac{2\sqrt{2}}{3}\right)}{\left(\frac{1}{3}\right)} \\ =-2\sqrt{2} \end{gathered}$$

Use the values of $\sin t,\cos t$and $\tan t$to find $\csc t$and $\cot t$.

$$\begin{gathered} \csc t=\frac{1}{\sin t} \\ =\frac{1}{\left(-\frac{2\sqrt{2}}{3}\right)} \\ =-\frac{3}{2\sqrt{2}} \end{gathered}$$