# Finding the Exact Value of an Inverse Trigonometry Function

• Jan 21st 2013, 09:04 PM
ianm
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!

• Jan 21st 2013, 09:08 PM
Prove It
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!

\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*}