Question

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

$\cos t=-\frac{7}{25}$, terminal point of t is in Quadrant III

Short answer

The values of the trigonometric functions are $\mathbf{sin}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{24}}{\mathbf{25}}\mathbf{,}\mathbf{tan}\mathit{t}\mathbf{=}\frac{\mathbf{24}}{\mathbf{7}}\mathbf{,}\mathbf{csc}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{24}}{\mathbf{25}}\mathbf{,}$$\mathbf{sec}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{25}}{\mathbf{7}}$,and $\mathbf{cot}\mathit{t}\mathbf{=}\frac{\mathbf{7}}{\mathbf{24}}$.

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. 

Now use Pythagorean Identity ${\sin }^{2}t+{\cos }^{2}t=1$, substitute $-\frac{7}{25}$for $\cos t$and simplify for $\cos t$.

role="math" localid="1648467860437" $$\begin{gathered} {\sin }^{2}t+{\cos }^{2}t=1 \\ {\sin }^{2}t+{\left(-\frac{7}{25}\right)}^2=1 \\ {\sin }^{2}t+\frac{49}{625}-\frac{49}{625}=1-\frac{49}{625} \\ {\sin }^{2}t=\frac{\text{576}}{625} \\ \sin t=\pm \frac{24}{25} \end{gathered}$$

Now, $t$is given to be in quadrant III, and quadrant III, $\sin t<0$, therefore, $\sin t=-\frac{24}{25}$.

Now substitute $-\frac{24}{25}$for $\sin t$and $-\frac{7}{25}$for $\cos t$into reciprocal identity $\tan t=\frac{\sin t}{\cos t}$and simplify for $\tan t$.

$$\begin{gathered} \tan t=\frac{\sin t}{\cos t} \\ =\frac{\left(-\frac{24}{25}\right)}{\left(-\frac{7}{25}\right)} \\ =\frac{24}{7} \end{gathered}$$

Step 3. Find other trigonometric functions.

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

role="math" localid="1648468120921" $$\begin{gathered} \csc t=\frac{1}{\sin t} \\ =\frac{1}{\left(-\frac{24}{25}\right)} \\ =-\frac{25}{24} \end{gathered}$$

$$\begin{gathered} \sec t=\frac{1}{\cos t} \\ =\frac{1}{\left(-\frac{7}{25}\right)} \\ =-\frac{25}{7} \end{gathered}$$

$$\begin{gathered} \cot t=\frac{1}{\tan t} \\ =\frac{1}{\left(\frac{24}{7}\right)} \\ =\frac{7}{24} \end{gathered}$$