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Math Help - Proving trig equation?

  1. #1
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    Proving trig equation?

    How would I prove the following.

    arcsin(4/5)+arctan(3/4)=pi/2

    Any help is appreciated.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Proving trig equation?

    You could draw a 3-4-5 right triangle and:

    a) find that \arcsin \left( \frac{4}{5} \right)= \text{arccot} \left( \frac{3}{4} \right)

    b) find that \arctan \left( \frac{3}{4} \right)= \arccos \left( \frac{4}{5} \right)

    Then use the fact that the sum of two complementary inverse trig functions having the same argument is always \frac{\pi}{2}.
    Last edited by MarkFL; December 13th 2012 at 06:43 PM.
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  3. #3
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    Re: Proving trig equation?

    I see why is it that the sum of two complementary inverse trig functions is pi/2?
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  4. #4
    MHF Contributor MarkFL's Avatar
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    Re: Proving trig equation?

    Because by definition, they represent complementary angles.
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  5. #5
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    Re: Proving trig equation?

    I see thanks.

    I have a similar question.

    It is

    arcsin(3/5)+arcsin(15/17)=(-13/85)

    How would I explain this problem. I know sin(a+b)=sinacosb+cosasinb.

    But I am not sure what to do.
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  6. #6
    MHF Contributor MarkFL's Avatar
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    Re: Proving trig equation?

    What you mean is:

    \arcsin\left(\frac{3}{5} \right)+\arcsin\left(\frac{15}{17} \right)=\arccos\left(-\frac{13}{85} \right)

    There are two ways we could proceed:

    a) Take the sine of both sides:

    \sin\left(\arcsin\left(\frac{3}{5} \right)+\arcsin\left(\frac{15}{17} \right) \right)=\sin\left(\arccos\left(-\frac{13}{85} \right) \right)

    \sin\left(\arcsin\left(\frac{3}{5} \right) \right)\cos\left(\arcsin\left(\frac{15}{17} \right) \right)+\cos\left(\arcsin\left(\frac{3}{5} \right) \right)\sin\left(\arcsin\left(\frac{15}{17} \right) \right)=\frac{\sqrt{85^2-(-13)^2}}{85}

    \frac{3}{5}\cdot\frac{\sqrt{17^2-15^2}}{17}+\frac{\sqrt{5^2-3^2}}{5}\cdot\frac{15}{17}=\frac{84}{85}

    \frac{3}{5}\cdot\frac{8}{17}+\frac{4}{5}\cdot\frac  {15}{17}=\frac{84}{85}

    \frac{24+60}{85}=\frac{84}{85}

    \frac{84}{85}=\frac{84}{85}

    b) Take the cosine of both sides:

    See if you can do this. Post your working, please.
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