Q.86 - Precalculus Enhanced with Graphing Utilities

Question

use properties of the trigonometric functions to find the exact value of each expression. Do not use a calculator: 

$\sec \left(-\frac{\pi }{18}\right)\cos \left(\frac{37\pi }{18}\right)$

Step-by-Step Solution

Answer

The exact value of $\sec \left(-\frac{\pi }{18}\right)\cos \left(\frac{37\pi }{18}\right)$ is $1$

Step 1: Given information

We are given a expression $\sec \left(-\frac{\pi }{18}\right)\cos \left(\frac{37\pi }{18}\right)$

Step 2: Use the periodicity of trigonometric function cos 2 π + θ = cos θ

We get

$$\begin{gathered} \sec \left(-\frac{\pi }{18}\right)\cos \left(\frac{37\pi }{18}\right) \\ \sec \left(\frac{\pi }{18}\right)\cos \left(2\pi +\frac{\pi }{18}\right) \\ \sec \left(\frac{\pi }{18}\right)\cos \left(\frac{\pi }{18}\right) \\ =\sec \left(\frac{\pi }{18}\right)\frac{1}{\sec \left({\displaystyle \frac{\pi }{18}}\right)} \\ =1 \end{gathered}$$

Step 3: Conclusion

The exact value of $\sec \left(\frac{\pi }{18}\right)\cos \left(\frac{37\pi }{18}\right) is 1.$