Q.85 - 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:  

$\sin \left(-\frac{\pi }{12}\right)\csc \left(\frac{25\pi }{12}\right)$

Step-by-Step Solution

Answer

The exact value of $\sin \left(-\frac{\pi }{12}\right)\csc \left(\frac{25\pi }{12}\right) is -1$

Step 1: Given information

We are given a trigonometric equation $\sin \left(-\frac{\pi }{12}\right)\csc \left(\frac{25\pi }{12}\right)$

Step 2: use the periodicity of trigonometric function csc θ + 2 π = csc θ

We get

$$\begin{gathered} \sin \left(-\frac{\pi }{12}\right)\csc \left(\frac{25\pi }{12}\right) \\ -\sin \left(\frac{\pi }{12}\right)\csc \left(2\pi +\frac{\pi }{12}\right) \\ -\sin \left(\frac{\pi }{12}\right)\csc \left(\frac{\pi }{12}\right) \\ -\sin \left(\frac{\pi }{12}\right)\frac{1}{\sin \left({\displaystyle \frac{\pi }{12}}\right)} \\ =-1 \end{gathered}$$

Step 3: Conclusion

The exact value of $\sin \left(-\frac{\pi }{12}\right)\csc \left(\frac{25\pi }{12}\right) is -1$