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Math Help - Trigonometric equation?

  1. #1
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    Trigonometric equation?

    If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?

    I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by D7236 View Post
    If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?

    I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?

    Thanks in advance.
    You can use identity \displaystyle \cos(\theta + \pi) = -\cos(\theta), to start out.

    Edit: Well I suppose it should be \displaystyle \cos(\theta - \pi) = -\cos(\theta), same idea.
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  3. #3
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    You should know

    \sin{(\alpha \pm \beta)} = \sin{\alpha}\cos{\beta} \pm \cos{\alpha}\sin{\beta}

    and

    \cos{(\alpha \pm \beta)} = \cos{\alpha}\cos{\beta} \mp \sin{\alpha}\sin{\beta}.


    So in your case

    3\sin{\left(x + \frac{\pi}{3}\right)} = 2\cos{\left(x - \frac{2\pi}{3}\right)}

    3\left(\sin{x}\cos{\frac{\pi}{3}} + \cos{x}\sin{\frac{\pi}{3}}\right) = 2\left(\cos{x}\cos{\frac{2\pi}{3}} + \sin{x}\sin{\frac{2\pi}{3}}\right)

    3\left(\frac{1}{2}\sin{x} + \frac{\sqrt{3}}{2}\cos{x}\right) = 2\left(-\frac{1}{2}\cos{x} + \frac{\sqrt{3}}{2}\sin{x}\right)

    \frac{3}{2}\sin{x} + \frac{3\sqrt{3}}{2}\cos{x} = -\cos{x} + \sqrt{3}\sin{x}

    \left(\frac{3}{2} - \sqrt{3}\right)\sin{x} = \left(-1 - \frac{3\sqrt{3}}{2}\right)\cos{x}

    \left(\frac{3 - 2\sqrt{3}}{2}\right)\sin{x} = -\left(\frac{2 + 3\sqrt{3}}{2}\right)\cos{x}

    \frac{\sin{x}}{\cos{x}} = \frac{-\frac{2 + 3\sqrt{3}}{2}}{\phantom{-}\frac{3 - 2\sqrt{3}}{2}}

    \tan{x} = -\frac{2 + 3\sqrt{3}}{3 - 2\sqrt{3}}

    \tan{x} = -\frac{(2 + 3\sqrt{3})(3 + 2\sqrt{3})}{(3 - 2\sqrt{3})(3 + 2\sqrt{3})}

    \tan{x} = -\frac{6 + 4\sqrt{3} + 9\sqrt{3} + 18}{9 - 12}

    \tan{x} = \frac{-(24 + 13\sqrt{3})}{-3}

    \tan{x} = \frac{24 + 13\sqrt{3}}{3}.
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  4. #4
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    Quote Originally Posted by undefined View Post
    You can use identity \displaystyle \cos(\theta + \pi) = -\cos(\theta), to start out.

    Edit: Well I suppose it should be \displaystyle \cos(\theta - \pi) = -\cos(\theta), same idea.
    Yes, we coud let \theta = x + \frac{\pi}{3}, but without the angle sum and difference identities we still can't reduce this to \tan{x}. So refer to my post above
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  5. #5
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    Quote Originally Posted by D7236 View Post
    If 3sin(x + pi/3) = 2cos(x - 2pi/3), find tanx?

    I started doing it using the sum and difference formulas but didn't end up getting the correct answer. Could someone show me how to do it using the sum and difference formulas?

    Thanks in advance.
    Since Prove It posted a full solution, here's another way.

    3\sin\left(x + \dfrac{\pi}{3}\right) = 2\cos\left(x - \dfrac{2\pi}{3}\right) = -2 \cos\left(x + \dfrac{\pi}{3}\right)

    \tan\left(x+\dfrac{\pi}{3} \right) = \dfrac{-2}{3}

    Use

    \displaystyle \tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}

    if you know it, otherwise use sin and cos angle sum like Prove It.

    Here

    \dfrac{-2}{3} = \dfrac{\tan x+\sqrt{3}}{1-\sqrt{3}\tan x}

    -2(1-\sqrt{3}\tan x) = 3(\tan x+\sqrt{3})

    -2+2\sqrt{3}\tan x = 3\tan x+3\sqrt{3}

    (2\sqrt{3}-3)\tan x = 3\sqrt{3}+2

    \tan x = \dfrac{2+3\sqrt{3}}{-3+2\sqrt{3}}

    \tan x = \dfrac{(2+3\sqrt{3})(-3-2\sqrt{3})}{(-3+2\sqrt{3})(-3-2\sqrt{3})}

    \tan x =  \dfrac{24+13\sqrt{3}}{3}

    Quote Originally Posted by Prove It View Post
    Yes, we coud let \theta = x + \frac{\pi}{3}, but without the angle sum and difference identities we still can't reduce this to \tan{x}. So refer to my post above
    I was busy typing and didn't see your recent post until just now. I think your way is a bit easier than mine, anyway.
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