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Math Help - sec^2(x)=4tan(x)

  1. #1
    Member Furyan's Avatar
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    sec^2(x)=4tan(x)

    Hello

    The question is solve on the interval -\pi\leq x \leq\pi

    \sec^2(x) = 4\tan(x)

    using the identity \sec^2(x) = \tan^2(x) +1

    I get:

    \tan^2(x) - 4\tan(x) +1 = 0

    I can't factor that, so I completed the square and got.

    \tan(x) = 2\pm\sqrt{3}

    It looks like that will give me the solutions, but I've never completed the square on a trigonometric function before and just wanted to check that it's the right way to proceed.

    Thank you.
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: sec^2(x)=4tan(x)

    Yes, what you did is the way to go.
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  3. #3
    Member Furyan's Avatar
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    Re: sec^2(x)=4tan(x)

    Quote Originally Posted by MarkFL2 View Post
    Yes, what you did is the way to go.
    Thank you MarkFL2,

    That's good to know. I'll keep going then .
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    Forum Admin topsquark's Avatar
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    Re: sec^2(x)=4tan(x)

    If you are solving for x there is another useful identity: the double angle formula.

    sec^2(x) = 4~tan(x)

    \frac{1}{cos^2(x)} = 4 \cdot \frac{sin(x)}{cos(x)}

    As long as we agree that cos(x) is not zero we may cancel one of the cosine terms:
    \frac{1}{cos(x)} = 4 \cdot sin(x)

    1 = 4 \cdot sin(x)~cos(x)

    Now, 2 sin(x) cos(x) = sin(2x). Thus

    1 = 2 \cdot sin(2x)

    And go from there. I get that
    x = \{ \frac{\pi}{12},~\frac{5 \pi}{12} \}
    Last edited by topsquark; November 23rd 2012 at 06:00 PM.
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  5. #5
    Member Furyan's Avatar
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    Re: sec^2(x)=4tan(x)

    Thank you topsquark,

    That's very helpful.

    I get the same solutions both ways. I think I need to know the double angle formula and it's really useful to know how to apply in this case
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  6. #6
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    Re: sec^2(x)=4tan(x)

    Just wondering, how do you complete the square on a trigonometric function? In Furyan's example  \tan^2(x) - 4\tan(x) +1 = 0 , half of b would be 2, and then squared is 4, but how do you condense  tan^2(x) -4tan(x)+5 ?
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  7. #7
    MHF Contributor MarkFL's Avatar
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    Re: sec^2(x)=4tan(x)

    You could write:

    \tan^2(x)-4\tan(x)=-1

    Add 4 to both sides:

    \tan^2(x)-4\tan(x)+4=3

    (\tan(x)-2)^2=3

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

    \tan(x)=2\pm\sqrt{3}
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  8. #8
    Forum Admin topsquark's Avatar
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    Re: sec^2(x)=4tan(x)

    Quote Originally Posted by AZach View Post
    Just wondering, how do you complete the square on a trigonometric function? In Furyan's example  \tan^2(x) - 4\tan(x) +1 = 0 , half of b would be 2, and then squared is 4, but how do you condense  tan^2(x) -4tan(x)+5 ?
    Perhaps it would be easier for you to define y = tan(x). Then we have
    y^2 - 4y + 1 = 0

    And of course once you get the solution for y then you plug back the tan(x) = y.

    -Dan
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