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Math Help - Finding the sine, cos of common angles without calculator

  1. #1
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    Finding the sine, cos of common angles without calculator

    Hi,

    I have heard its possible to find the sine and cosine of some common angles without calculator.

    Can anyone explain how please?

    eg.

    cos pi/4 = 1/√2
    sin pi/4 = 1/√2

    cos 5pi/4 = -1/√2

    cos pi/3 = 1/2

    sin pi/3 = √3/2


    Thanks in advance
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  2. #2
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    Re: Finding the sine, cos of common angles without calculator

    Look at the thread when tanx = 1
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  3. #3
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    Re: Finding the sine, cos of common angles without calculator

    Hello, ScorpFire!

    I have heard its possible to find the sine and cosine of some common angles without calculator.

    Can anyone explain how please?

    For example: . \begin{array}{cccccccccccc}\sin\frac{\pi}{4} \:=\:\frac{1}{\sqrt{2}} && \sin\frac{\pi}{3} \:=\:\frac{\sqrt{3}}{2} && \sin\frac{\pi}{6} \:=\:\frac{1}{2} \\ \\[-3mm] \cos\frac{\pi}{4} \:=\:\frac{1}{\sqrt{2}} && \cos\frac{\pi}{3} \:=\:\frac{1}{2} && \cos\frac{\pi}{6} \:=\:\frac{\sqrt{3}}{2} \end{array}






    \text{For }\theta = \tfrac{\pi}{4}\:(45^o), consider an isosceles right triangle.
    Code:
          *
          * *
          *   *  h
        1 *     *
          *       *
          *      45 *
          *  *  *  *  *
                1
    Let the equal sides equal 1.
    Pythagorus says the hypotenuse is \sqrt{2}.

    We have: . \begin{Bmatrix} opp &=& 1 \\ adj &=& 1 \\ hyp &=& \sqrt{2}\end{Bmatrix}
    And you can write the trig values of \tfrac{\pi}{4}

    Memorize "one, one, square-root-of-two".



    \text{For }\theta =\tfrac{\pi}{3}\:(60^o), consider an equilateral triangle
    . . with side length 2. .Draw an altitude.
    Code:
                *
               *|*
              * | *
           2 *  |y *
            *   |   *
           * 60 |    *
          *  *  *  *  *
          :  1  :  1  :
    We have: . adj = 1,\:hyp = 2
    Pythagorus says: opp = \sqrt{3}
    And you can write the trig values for \tfrac{\pi}{3}



    For \theta = \tfrac{\pi}{6}\;(30^o), turn the above right triangle on its side.
    Code:
                      *
              2    *  *
                *     * 1
             * 30     *
          *  *  *_ *  *
                √3
    We have: . \begin{Bmatrix}opp &=& 1 \\ adj &=& \sqrt{3} \\ hyp &=& 2\end{Bmatrix}
    And you can write the trig values for \tfrac{\pi}{6}


    For both diagrams, memorize "one, two, square-root-of-three".

    Be careful! .Remember that:
    . . The shortest side, 1, is opposite the smallest angle, 30o.
    . . The longest side, 2, is opposite the largest angle, 90o.
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  4. #4
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    Re: Finding the sine, cos of common angles without calculator

    Take a point on the unit circle. The cosine of the corresponding angle is simply the x-coordinate (adjacent over hypotenuse). The sine of that angle is the y-coordinate. That's how I usually find sine/cosine in my head by just visualizing the unit circle.
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  5. #5
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    Re: Finding the sine, cos of common angles without calculator

    There are formulae for finding sin(A+B) and cos(A+B) etc... using these one can extend the values for other angles. For example, sin2A = 2sinAcosA, using which one can find sin(15 degrees) = sqrt(2 - sqrt(3))/2.

    Salahuddin
    Maths online
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