Question

Find the exact values of the six trigonometric functions of the given angle. If any are not defined, say "not defined." Do not use a calculator.

$5\pi$

Short answer

The exact values of the six trigonometric functions are$\sin \left(5\pi \right)=0$,$\cos \left(5\pi \right)=-1$,$\tan \left(5\pi \right)=0$,$\csc \left(5\pi \right)=\text{Not defined}$,$\sec \left(5\pi \right)=-1$and$\cot \left(5\pi \right)=\text{Not defined}$.

Step-by-step solution

Step 1. Determine a point corresponds to the given angle.

The given angle is $5\pi =4\pi +\pi =\pi$. This angle is multiple of $\pi$.

In a unit circle, the point $\left(-1,0\right)$corresponds to $\pi$.

Step 2. Determine the exact value of sine function.

The exact value of the sine function is:

$$\begin{gathered} \sin \left(5\pi \right)=y \\ \sin \left(5\pi \right)=0 \end{gathered}$$

Step 3. Determine the exact value of cosine function.

The exact value of the cosine function is:

$$\begin{gathered} \cos \left(5\pi \right)=x \\ \cos \left(5\pi \right)=-1 \end{gathered}$$

Step 4. Determine the exact value of tangent function.

The exact value of the tangent function is:

$\begin{array}{rcl}\tan \theta & =& \frac{\sin \theta }{\cos \theta }\\ \tan \left(5\pi \right)& =& \frac{0}{-1}\\ \tan \left(5\pi \right)& =& 0\end{array}$

Step 5. Determine the exact value of cosecant function.

The exact value of the cosecant function is:

$\begin{array}{rcl}\csc \theta & =& \frac{1}{\sin \theta }\\ \csc \left(5\pi \right)& =& \frac{1}{0}\\ \csc \left(5\pi \right)& =& \text{Not defined}\end{array}$

Step 6. Determine the exact value of secant function.

The exact value of the secant function is:

$\begin{array}{rcl}\sec \theta & =& \frac{1}{\cos \theta }\\ \sec \left(5\pi \right)& =& \frac{1}{-1}\\ \sec \left(5\pi \right)& =& -1\end{array}$

Step 7. Determine the exact value of cotangent function.

The exact value of the cotangent function is:

$\begin{array}{rcl}\cot \theta & =& \frac{1}{\tan \theta }\\ \cot \left(5\pi \right)& =& \frac{1}{0}\\ \cot \left(5\pi \right)& =& \text{Not defined}\end{array}$