Question

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

$\mathbf{sin}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{1}}{\mathbf{4}}\mathbf{,}\mathbf{sec}\mathbf{}\mathit{t}\mathbf{<}\mathbf{0}$

Short answer

The values of the trigonometric functions are $\mathbf{cos}\mathbf{}\mathit{t}\mathbf{=}\mathbf{-}\frac{\sqrt{\mathbf{15}}}{\mathbf{4}}$, $\mathbf{tan}\mathbf{}\mathit{t}\mathbf{=}\frac{\mathbf{1}}{\sqrt{\mathbf{15}}}$,$\mathbf{csc}\mathbf{}\mathit{t}\mathbf{=}\mathbf{-}\mathbf{4}$,$\mathbf{sec}\mathbf{}\mathit{t}\mathbf{=}\mathbf{-}\frac{\mathbf{4}}{\sqrt{\mathbf{15}}}$,and$\mathbf{cot}\mathbf{}\mathit{t}\mathbf{=}\sqrt{\mathbf{15}}$

Step-by-step solution

Step 1. State the Trigonometric identities.

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

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. Find cos t and tan t.

Now use Pythagorean Identity ${\sin }^{2}t+{\cos }^{2}t=1$, substitute $-\frac{1}{4}$for localid="1650451102652" $\sin t$and simplify for $\cos t$.

$\begin{array}{l}{\sin }^{2}t+{\cos }^{2}t=1\\ {\left(-\frac{1}{4}\right)}^2+{\cos }^{2}t=1\\ \frac{1}{16}+{\cos }^{2}t-\frac{1}{16}=1-\frac{1}{16}\\ {\cos }^{2}t=\frac{15}{16}\\ \cos t=\pm \frac{\sqrt{15}}{4}\end{array}$

Since, $\sec t<0$, therefore,$\cos t<0$because$\sec t=\frac{1}{\cos {\displaystyle }t}$, so,$\cos t=-\frac{\sqrt{15}}{4}$.

Now substitute $-\frac{1}{4}$for $\sin t$and $-\frac{\sqrt{15}}{4}$for $\cos t$into reciprocal identity $\tan t=\frac{\sin {\displaystyle }t}{\cos {\displaystyle }t}$and simplify for $\tan t$.

$\begin{array}{l}\tan t=\frac{\sin t}{\cos t}\\ =\frac{\left(-\frac{1}{4}\right)}{\left(-\frac{\sqrt{15}}{4}\right)}\\ =\frac{1}{\sqrt{15}}\end{array}$

Step 3. Find other trigonometric functions.

Use the values of and to find and .

$\begin{array}{l}\csc t=\frac{1}{\sin t}\\ =\frac{1}{\left(-\frac{1}{4}\right)}\\ =-4\end{array}$

$\begin{array}{l}\sec t=\frac{1}{\cos t}\\ =\frac{1}{\left(-\frac{\sqrt{15}}{4}\right)}\\ =-\frac{4}{\sqrt{15}}\end{array}$

$\begin{array}{l}\cot t=\frac{1}{\tan t}\\ =\frac{1}{\left(\frac{1}{\sqrt{15}}\right)}\\ =\sqrt{15}\end{array}$