Question

Use the even–odd properties to find the exact value of each expression. Do not use a calculator.

$\csc (-\frac{\mathrm{\pi }}{4})$

Short answer

The exact value of trigonometry expression is$-\sqrt{2}.$

Step-by-step solution

Step 1. Given Information

Given that the trigonometry expression is $\csc (-\frac{\mathrm{\pi }}{4}).$To find the exact value of trigonometry expression using even odd property.

Step 2. Even odd property

Cosecant is an odd function and it can be denoted as,

$$\begin{gathered} \csc (-\mathrm{\theta })=-\mathrm{csc\theta } \\ =-\frac{1}{\mathrm{sin\theta }} \\ =-\frac{1}{\mathrm{y}} \end{gathered}$$

The point in which on unit circle responds the angle can be substituted.

$$\begin{gathered} \csc (-\frac{\mathrm{\pi }}{4})=-\csc \frac{\mathrm{\pi }}{4} \\ =\frac{1}{-\mathrm{y}} \\ =\frac{1}{-{\displaystyle \frac{\sqrt{2}}{2}}} \\ \csc (-\frac{\mathrm{\pi }}{4})=-\sqrt{2} \end{gathered}$$