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Math Help - sines

  1. #1
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    sines

    I need help proving these 2 identities.

    1. If x = 18 degrees, prove that sin2x = cos3x. Find the exact values of sinx and cosx

    2. If 2sin(x - y) = sin(x + y), prove that tanx = 3tan y

    Please help!
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Zetterbergx40 View Post
    2. If 2sin(x - y) = sin(x + y), prove that tanx = 3tan y
    2sin(x - y) = sin(x + y)

    2sin(x)cos(y) - 2sin(y)cos(x) = sin(x)cos(y) + sin(y)cos(x)

    sin(x)cos(y) = 3sin(y)cos(x) <-- Divide both sides by cos(x)cos(y)

    tan(x) = 3tan(y)

    -Dan
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  3. #3
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by Zetterbergx40 View Post
    1. If x = 18 degrees, prove that sin2x = cos3x. Find the exact values of sinx and cosx
    I can't think of a way to help you on the first part, but for finding the exact value of sine and cosine:
    sin(2x) = cos(3x)

    2sin(x)cos(x) = -4sin^2(x) cos(x) + cos(x) <-- Divide both sides by cos(x)

    2sin(x) = -4sin^2(x) + 1

    4sin^2(x) + 2sin(x) - 1 = 0

    Using the quadratic formula:
    sin(x) = \frac{-1 \pm \sqrt{5}}{4}

    Note that both solutions are acceptable, indicating that there are actually 4 solutions for x in 0 \leq x < 360. Obviously the sin(18) will correspond to the positive solution.

    From the sine equation we can find cosine:
    cos(x) = \pm \sqrt{1 - sin^2(x)}
    where the \pm indicates which quadrant the angle is in. (So for 18 degrees, pick + because cosine is positive in the first quadrant.)

    -Dan
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  4. #4
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    Thanks Dan

    Thanks alot, for the first reply, can you tell me what the solution would be in LS = RS form? Thanks
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  5. #5
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    Hello, Zetterbergx40!

    1. If x = 18^o, prove that: . \sin2x \:= \:\cos3x

    We are asked to prove that: . \sin36^o \:=\:\cos54^o

    The two angles are complementary and \sin\theta \:=\:\cos(90^o - \theta)


    Or look at this triangle:
    Code:
                  *
                 /|
                / |
               /  |
              /36|
           c /    | b
            /     |
           /      |
          /       |
         / 54    |
        *---------*
             a
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