Finding the Exact Value of an Inverse Trigonometry Function

Hello,

I want to find the exact value of $\displaystyle csc(-\frac{5\pi}{3}) $

I know that $\displaystyle csc(\theta)=\frac{1}{sin(\theta)}$, and the answer to the question is $\displaystyle \frac{2\sqrt{3}}{3} $, but don't know how this works!

Thank you for your answers!

Re: Finding the Exact Value of an Inverse Trigonometry Function

Quote:

Originally Posted by

**ianm** Hello,

I want to find the exact value of $\displaystyle csc(-\frac{5\pi}{3}) $

I know that $\displaystyle csc(\theta)=\frac{1}{sin(\theta)}$, and the answer to the question is $\displaystyle \frac{2\sqrt{3}}{3} $, but don't know how this works!

Thank you for your answers!

$\displaystyle \displaystyle \begin{align*} \csc{\left( -\frac{5\pi}{3} \right)} &= \frac{1}{\sin{\left( -\frac{5\pi}{3} \right)}} \\ &= \frac{1}{\sin{\left( \frac{\pi}{3} \right)}} \textrm{ since } \sin{\left( \theta + 2\pi \right)} = \sin{(\theta)} \\ &= \frac{1}{\frac{\sqrt{3}}{2}} \\ &= \frac{2}{\sqrt{3}} \\ &= \frac{2\sqrt{3}}{3} \end{align*}$