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Math Help - Find exact values of inverse trig functions.

  1. #1
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    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.
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  2. #2
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    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
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  3. #3
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    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.
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  4. #4
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    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.
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  5. #5
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    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.
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  6. #6
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    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.
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    Re: Find exact values of inverse trig functions.

    Quote Originally Posted by FatimaA View Post
    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.
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