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Math Help - Trigonometric equations

  1. #1
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    Exclamation Trigonometric equations

    Solve the following equation for angles between 0 degrees and 360 degrees inclusive.

    2 sin(square) x 5 sin x cos x = 3 cos(square) x
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  2. #2
    Bar0n janvdl's Avatar
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    Quote Originally Posted by Ilsa View Post
    Solve the following equation for angles between 0 degrees and 360 degrees inclusive.

    2 sin(square) x – 5 sin x cos x = 3 cos(square) x
    2 sin^2 x - 5 sin x cos x - 3 cos^2 x = 0

    What if we let sinx = u and cos x = v ?

    2u^2 - 5uv - 3v^2 = 0

    Can you continue?
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  3. #3
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    do I solve it further like this:
    2u (square) - 6uv +uv - 3v (square)
    and then factorise?

    After factorisation, I got:
    sin x - 3 cos x = 0
    or 2 sinx + cos x = 0

    If I substitute, short equations like 7 cos x = 0 are coming up, through which it is not possible to deduce an answer mathematically.
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  4. #4
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    the answers are supposed to be: 71.6 degrees, 153.4 degrees, 251.6 degrees and 333.4 degrees.
    However, I am not able to solve the equation and attain those answers.
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  5. #5
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    Hello, Ilsa!

    You're off to a good start . . .


    \text{Do I solve it further like this?}

    . . 2u^2 - 5uv - 3v^2 \:=\:0\quad\Rightarrow\quad (u - 3v)(2u + v) \:=\:0

    \text{Then: }\;\begin{Bmatrix}\sin x - 3\cos x &=& 0 & [1] \\<br />
2\sin x + \cos x &=& 0 & [2] \end{Bmatrix}

    From [1] we have:
    . . \sin x \:=\:3\cos x \quad\Rightarrow\quad \dfrac{\sin x}{\cos x}\:=\:3 \quad\Rightarrow\quad \tan x \:=\:3

    Hence: . x \:=\:\arctan 3 \quad\Rightarrow\quad \boxed{x\:\approx\: 71.6^o,\:251.6^o}


    From [2] we have:
    . . 2\sin x \:=\:-\cos x \quad\Rightarrow\quad \dfrac{\sin x}{\cos x} \:=\:\text{-}\frac{1}{2} \quad\Rightarrow\quad \tan x \:=\:\text{-}\frac{1}{2}

    Hence: . x \:=\:\arctan(\text{-}0.5) \quad\Rightarrow\quad \boxed{x \:\approx\: 153.4^o,\:333.4^o}

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  6. #6
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    Thankyou, Mr. Soroban.
    The solution you gave really helped.
    I was substituting sinx = 3cosx into the next equation, which turned out be wrong.
    Thankyou for your help!
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