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Math Help - calculating trig. ratios w/ out a calculator

  1. #1
    Junior Member jonnygill's Avatar
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    calculating trig. ratios w/ out a calculator

    Hello,

    So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.

    Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
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  2. #2
    Super Member TheChaz's Avatar
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    Quote Originally Posted by jonnygill View Post
    Hello,

    So i know how to construct a table for 0, 30, 45, 60, and 90 degree angles (and the respective radians) that shows the trig. ratios for these degree/radian values. If i am asked to determine the trig. ratio for 30 degrees without a calculator i'm cool w/ that. What I cannot always do is determine trig ratios for csc 120 (or another degree value other than the values i construct the table for). I figured I could just take the inverse of 2 times the sin of 60 degrees, but this didn't result in the correct answer as determined w/ the use of a calculator. I would think this would always work (that is, adding up the trig ratios for an angle to determine a greater angle that is a multiple of the smaller angles). Sometimes it works, but sometimes it doesn't.

    Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
    Yes. Lookup "unit circle".
    The format of the points around the circle is (cosx, sinx). So find the point corresponding to 120 degrees, find the sin value (√3/2) and flip it.
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  3. #3
    Junior Member jonnygill's Avatar
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    thank you.

    but what if such a chart is not available?
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  4. #4
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jonnygill View Post
    Does anyone have a suggestion for determining trig ratios w/ out a calculator. (e.g. csc 120).
    \displaystyle csc(120) = \frac{1}{sin(120)} = \frac{1}{sin(2 \cdot 60 )} = \frac{1}{2~sin(60)~cos(60)}

    \displaystyle = \frac{1}{2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}

    \displaystyle = \frac{2}{\sqrt{3}}

    -Dan
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  5. #5
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    You have to commit to memory the special triangles for 30,45 and 60 and then know the sign of each function corresponding to each quadrant. Have you seen CAST?
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  6. #6
    Junior Member jonnygill's Avatar
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    Quote Originally Posted by topsquark View Post
    \displaystyle csc(120) = \frac{1}{sin(120)} = \frac{1}{sin(2 \cdot 60 )} = \frac{1}{2~sin(60)~cos(60)}

    \displaystyle = \frac{1}{2 \cdot \frac{\sqrt{3}}{2} \cdot \frac{1}{2}}

    \displaystyle = \frac{2}{\sqrt{3}}

    -Dan
    Where did cos come from? how does one know when to use cos instead of sin in problems like this?
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  7. #7
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by jonnygill View Post
    Where did cos come from? how does one know when to use cos instead of sin in problems like this?
    I used the identity sin(2 \theta) = 2~sin( \theta )~cos( \theta ). It's one of the double angle formulas.

    -Dan
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  8. #8
    MHF Contributor
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    Better yet, note that \displaystyle \sin{\left(120^{\circ}\right)} = \sin{\left(180^{\circ} - 60^{\circ}\right)}

    \displaystyle = \sin{\left(60^{\circ}\right)}

    \displaystyle = \frac{\sqrt{3}}{2}.
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  9. #9
    MHF Contributor
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    Quote Originally Posted by jonnygill View Post
    thank you.

    but what if such a chart is not available?
    live it ... learn it ... luv it.

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