# Thread: Find exact values of inverse trig functions.

1. ## Find exact values of inverse trig functions.

Find the exact value of the following. Draw a picture to help explain your answer.

sin^(-1)(sin(4π/3))

This is what I did:

sin^(-1)(sin(4π/3)) = sin^(-1)(-√3/2) = -π/3

Is this correct? I'm still not clear on how to determine which quadrant it should be in, but I guessed it would be quadrant 4 because the sine is negative.

2. ## Re: Find exact values of inverse trig functions.

Sine is negative in the 3rd and 4th quadrants.

Also note that if f(x) is a function and f-1(x) is its inverse, then f-1(f(a))=a

3. ## Re: Find exact values of inverse trig functions.

Hi, thanks for your help. I understand that sine is negative in the 3rd and 4th quadrants. My question is what quadrant would π/3 be in? Or in other words, would a be positive or negative if f^-1(f(-a))?

Thanks again.

4. ## Re: Find exact values of inverse trig functions.

Do you not know what a "quadrant" is? A quadrant is 1/4 of a the plane or 1/4 of a circle so the "first quadrant" is from 0 to $\pi/2$, the "second quadrant" is from $\pi/2$ to $\pi$, the "third quadrant" is from $\pi$ to $3\pi/2$ and the "fourth quadrant" is from $3\pi/2$ to $2\pi$.

5. ## Re: Find exact values of inverse trig functions.

Thank you. I know what a quadrant is. I'm not a math major, but I have the courage to come here and ask about things I'm ignorant about because the people here are more knowledgeable about it than I am. I come here and very respectfully ask for help. I am here to learn, after all. I would appreciate being treated with respect as well.

6. ## Re: Find exact values of inverse trig functions.

You need to remember that the \displaystyle \begin{align*} \arcsin{(x)} \end{align*} function is DEFINED so that you get an answer in the region \displaystyle \begin{align*} \left[ -\frac{\pi}{2} , \frac{\pi}{2} \right] \end{align*}. So even though your original angle was \displaystyle \begin{align*} \frac{4\pi}{3} \end{align*}, you are correct that you will get an answer of \displaystyle \begin{align*} -\frac{\pi}{3} \end{align*}, due to the original restriction made so that the function is one-to-one.

7. ## Re: Find exact values of inverse trig functions.

Originally Posted by FatimaA
Thank you. I know what a quadrant is. I'm not a math major, but I have the courage to come here and ask about things I'm ignorant about because the people here are more knowledgeable about it than I am. I come here and very respectfully ask for help. I am here to learn, after all. I would appreciate being treated with respect as well.
I can completely understand your frustration and even your thinking disrespect was given.
I can assure you that no disrespect was meant. We assume that you have certain basic knowledge.

In this case, the function $\arcsin$ has a domain of $[-1,1]$.

But more importantly, the range is $\left[ {\frac{{ - \pi }}{2},\frac{\pi }{2}} \right]$

If you know those two facts then this whole thread is mute. In other words, knowing the basics answers most questions.