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Math Help - Calculating the Exact Value of Trig Functions

  1. #1
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    Calculating the Exact Value of Trig Functions

    Hi Guys,

    I have another question:calculate the exact value of the following expression:

    <br />
tan(arcsin(\frac{2}{3}))<br />
    I don't know how to calculate the exact value of arcsin(\frac{2}{3})

    Can anyone help?
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  2. #2
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    Quote Originally Posted by spycrab View Post
    Hi Guys,

    I have another question:calculate the exact value of the following expression:

    <br />
tan(arcsin(\frac{2}{3}))<br />
    I don't know how to calculate the exact value of arcsin(\frac{2}{3})

    Can anyone help?
    1. Draw a sketch. (see attachment)

    2. Use proportions:

    \dfrac{\frac23}1 = \dfrac{y}{\sqrt{1+y^2}}

    3. Since y = \tan\left(\arcsin\left(\frac23}\right) \right) solve the equation at 2. for y.

    4. I've got y = \tan\left(\arcsin\left(\frac23}\right) \right) = \dfrac25 \sqrt{5}
    Attached Thumbnails Attached Thumbnails Calculating the Exact Value of Trig Functions-exacttan.png  
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  3. #3
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    thanks!
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  4. #4
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    Hello, spycrab!

    Calculate the exact value of: . \tan\left[\arcsin (\frac{2}{3}\right]

    Let \theta = \arcsin(\frac{2}{3})\right

    Then: . \sin\theta \:=\:\frac{2}{3} \:=\:\frac{opp}{hyp}


    That is, \theta is in a right triangle with opp = 2,\;hyp = 3

    Pythagorus tells us that: . adj \,=\,\sqrt{5}

    Hence: . \tan\theta \:=\: \frac{opp}{adj} \:=\:\frac{2}{\sqrt{5}}


    Therefore: . \tan\left[\arcsin(\frac{2}{3})\right] \;=\;\dfrac{2}{\sqrt{5}}
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  5. #5
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    thanks
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