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Math Help - Given tan (theta) what is sin (theta)

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    Given tan (theta) what is sin (theta)

    if pi/2 ≤ theta ≤ 3pi/2 and tanθ = -4/3
    Determine the exact value of sinθ

    if the answer was something I could use the special triangles to solve It wouldnt be so hard ie(1/2, root3/2 ....)

    I could maybe draw it out and solve, but how do you do something like this?
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    Quote Originally Posted by brentwoodbc View Post
    if pi/2 ≤ theta ≤ 3pi/2 and tanθ = -4/3
    Determine the exact value of sinθ

    if the answer was something I could use the special triangles to solve It wouldnt be so hard ie(1/2, root3/2 ....)

    I could maybe draw it out and solve, but how do you do something like this?
    \frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2} means that you're in either the 2nd or 3rd quadrant. But tan is negative in the 3rd quadrant and the 4th quadrant. Therefore \theta lies in the second quadrant.

    Now draw a right-triangle with angle \theta such that the opposite side is 4 and the adjacent side is 3. Clearly the hypotenuse is 5. Then \sin \theta = \frac{4}{5} (because sine is positive in the 2nd quadrant).
    Last edited by mr fantastic; June 5th 2009 at 08:24 PM. Reason: Fixed a careless mistake
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    how do you know what quadrant it is in? how do you get theta from the triangle?
    Last edited by mr fantastic; June 5th 2009 at 08:28 PM. Reason: Edited because of an earlier mistake I made.
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    Quote Originally Posted by brentwoodbc View Post
    Thank you.

    + or - ? answers given are

    -4/5

    -3/5

    3/5

    4/5
    I edited my original post because I made a careless mistake.

    Quote Originally Posted by brentwoodbc View Post
    [snip] how do you get theta from the triangle?
    Use basic trigonometry to get the magnitude of \sin \theta. Then decide whether it's plus or minus.
    Last edited by mr fantastic; June 5th 2009 at 08:26 PM.
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    Im not allowed a calc. so how else would I find if it's plus/ neg.
    Last edited by mr fantastic; June 5th 2009 at 08:26 PM.
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    Quote Originally Posted by brentwoodbc View Post
    [snip]
    Im not allowed a calc. so how else would I find if it's plus/ neg.
    I assume that you're expected to be familiar with how the definition of sine (and cosine) relates to the unit circle. That's why I have refered to quadrants. You're expected to know what the sign of sine (and cosine) and tan is in each quadrant.
    Last edited by mr fantastic; June 5th 2009 at 08:27 PM.
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    Quote Originally Posted by mr fantastic View Post
    \frac{\pi}{2} \leq \theta \leq \frac{3 \pi}{2} means that you're in either the 2nd or 3rd quadrant. But tan is negative in the 3rd quadrant and the 4th quadrant. Therefore \theta lies in the second quadrant.

    Now draw a right-triangle with angle \theta such that the opposite side is 4 and the adjacent side is 3. Clearly the hypotenuse is 5. Then \sin \theta = \frac{4}{5} (because sine is positive in the 2nd quadrant).
    thanks that makes more sense, but isnt tan supposed to be negative?
    (tan(theta)=-4/3)
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    Quote Originally Posted by brentwoodbc View Post
    thanks that makes more sense, but isnt tan supposed to be negative?
    (tan(theta)=-4/3)
    In the second quadrant tan is negative and sine is positive. The triangle I used earlier gives you the size of \sin \theta. Then you examine the quadrant the angle is in to decide whether the answer is +ve or -ve.
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