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Math Help - Trigonometry Question

  1. #1
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    Trigonometry Question

    I figured I would post this in here and hopefully get somebody who likes trig to help me through this.


    Given that cos x = (3^.5) − 1, find the value of cos 2x in the form a +b(3^.5) ,
    where a and b are integers.

    Given that
    2 cos (y + 30)= 3 sin (y − 30),
    find the value of tan y in the form k 3 where k is a rational constant.


    a bit of background here is that I am looking to take my high school maths certificate and I found out that well... Im rubbish at trig!
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  2. #2
    MHF Contributor
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    Hello
    cosx\;=\;\sqrt{3}-1

    Using the formula
    cos(2x) = 2cos²x - 1

    cos(2x)\;=\;2(\sqrt{3}-1)^2-1
    cos(2x)\;=\;2(4-2\sqrt{3})-1
    cos(2x)\;=\;7-4\sqrt{3}
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  3. #3
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    Hello, GregoryT!

    1) Given that: . \cos x \:=\:\sqrt{3}-1,
    find the value of \cos 2x in the form a +b\sqrt{3}, where a\text{ and }b are integers.
    We are expected to know the identity: . \cos 2x \:=\:2\cos^2\!x - 1

    Then we have: . \cos2x \;=\;2(\sqrt{3}-1)^2 - 1

    . . =\;\;2(3 - 2\sqrt{3} + 1) - 1 \;\;=\;\;6 - 4\sqrt{3} + 2 - 1 \;\;=\;\;\boxed{7 -4\sqrt{3}}




    Given that: . 2\cos(y + 30)\:=\:3\sin(y - 30),

    find the value of \tan y . . . in the form k 3 where k is a rational constant. . ??

    We have: . 2\cos(y + 30) \:=\:3\sin(y-30)

    . . 2\bigg(\cos y \cos30 -\sin y\sin30\bigg) \;=\;3\bigg(\sin y\cos30 - \cos y\sin30\bigg)

    . . 2\bigg(\frac{\sqrt{3}}{2}\cos y - \frac{1}{2}\sin y\bigg) \;=\;3\bigg(\frac{\sqrt{3}}{2}\sin y - \frac{1}{2}\cos y\bigg)


    Multiply by 2: . 2\sqrt{3}\cos y - 2\sin y \;=\;3\sqrt{3}\sin y - 3\cos y

    . . 3\sqrt{3}\sin y + 2\sin y \;=\;2\sqrt{3}\cos y + 3\cos y

    . . (3\sqrt{3}+2)\sin y \;=\;(2\sqrt{3} + 3)\cos y

    . . \frac{\sin y}{\cos y} \;=\;\frac{2\sqrt{3}+3}{3\sqrt{3}+2}

    . . \tan y \;=\;\frac{2\sqrt{3}+3}{3\sqrt{3}+2} \cdot{\color{blue}\frac{3\sqrt{3}-2}{3\sqrt{3}-2} } \;=\;\frac{18 - 4\sqrt{3} + 9\sqrt(3) - 6}{27 - 4}


    Therefore: . \boxed{\tan y \;=\;\frac{12+5\sqrt{3}}{23}}

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  4. #4
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    Thanks Soroban and running gag, You have been invaluable help. Especially on that last one soroban! My formal Board thanks to both of you!
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  5. #5
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    Sorry, one more quick question, can you explain what is happening on the line immediately before the answer.
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