Q. 61 - Precalculus Enhanced with Graphing Utilities

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.

$\frac{5\pi }{2}$

Step-by-Step Solution

Answer

The exact values of the six trigonometric functions are $\sin \left(\frac{5\pi }{2}\right)=1$, $\cos \left(\frac{5\pi }{2}\right)=0$, $\tan \left(\frac{5\pi }{2}\right)=\text{Not defined}$, $\csc \left(\frac{5\pi }{2}\right)=1$, $\sec \left(\frac{5\pi }{2}\right)=\text{Not defined}$ and $\cot \left(\frac{5\pi }{2}\right)=0$.

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

The given angle is $\frac{5\pi }{2}=2\pi +\frac{\pi }{2}=\frac{\pi }{2}$. This angle is multiple of $\frac{\pi }{2}$.

In a unit circle, the point $\left(0,1\right)$ corresponds to $\frac{\pi }{2}$.

Step 2. Determine the exact value of sine function.

The exact value of the sine function is:

$$\begin{gathered} \sin \left(\frac{5\pi }{2}\right)=y \\ \sin \left(\frac{5\pi }{2}\right)=1 \end{gathered}$$

Step 3. Determine the exact value of cosine function.

The exact value of the cosine function is:

$$\begin{gathered} \cos \left(\frac{5\pi }{2}\right)=x \\ \cos \left(\frac{5\pi }{2}\right)=0 \end{gathered}$$

Step 4. Determine the exact value of tangent function.

The exact value of the tangent function is:

$\begin{array}{rcl}\tan \theta & =& \frac{\sin \theta }{\cos \theta }\\ \tan \left(\frac{5\pi }{2}\right)& =& \frac{1}{0}\\ \tan \left(\frac{5\pi }{2}\right)& =& \text{Not defined}\end{array}$

Step 5. Determine the exact value of cosecant function.

The exact value of the cosecant function is:

$\begin{array}{rcl}\csc \theta & =& \frac{1}{\sin \theta }\\ \csc \left(\frac{5\pi }{2}\right)& =& \frac{1}{1}\\ \csc \left(\frac{5\pi }{2}\right)& =& 1\end{array}$

Step 6. Determine the exact value of secant function.

The exact value of the secant function is:

$\begin{array}{rcl}\sec \theta & =& \frac{1}{\cos \theta }\\ \sec \left(\frac{5\pi }{2}\right)& =& \frac{1}{0}\\ \sec \left(\frac{5\pi }{2}\right)& =& \text{Not defined}\end{array}$

Step 7. Determine the exact value of cotangent function.

The exact value of the cotangent function is:

$\begin{array}{rcl}\cot \theta & =& \frac{\cos \theta }{\sin \theta }\\ \cot \left(\frac{5\pi }{2}\right)& =& \frac{0}{1}\\ \cot \left(\frac{5\pi }{2}\right)& =& 0\end{array}$