Question

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

$\csc (-\frac{\mathrm{\pi }}{3})$

Short answer

The exact value of trigonometric expression is$-\frac{2\sqrt{3}}{3}.$

Step-by-step solution

Step 1. Given Information

Given that the trigonometry expression is $\csc (-\frac{\mathrm{\pi }}{3}).$To find the exact value of the expression using even odd property.

Step 2. Even odd property

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

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

The point on the circle corresponds to the given angle.

$$\begin{gathered} \csc (-\frac{\mathrm{\pi }}{3})=-\csc \frac{\mathrm{\pi }}{3} \\ =\frac{1}{-\mathrm{y}} \\ =\frac{1}{-{\displaystyle \frac{\sqrt{3}}{2}}} \\ =-\frac{2\sqrt{3}}{3} \end{gathered}$$