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Math Help - Trig equation and identities

  1. #1
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    Trig equation and identities

    Sitting here studying for a final exam this week, and I am SOO bad at identities. Maybe I just don't know them well enough.
    Anyhow,

    Find the solutions to the following:
     sec^2x = 2secx

    I got this far...or maybe it's all wrong, I duno.
     1+tan^2x = 2secx * from a Pythagorean identity
     1+tan^2x = \frac{1}{2cosx} * from the reciprocal of sec?
     1+tan^2x (cos2x-sin2x) *cross multiplied and double angle formula? I feel this wasn't a good move, haha.

    Thanks in advanced!
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  2. #2
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by tiar View Post
    Sitting here studying for a final exam this week, and I am SOO bad at identities. Maybe I just don't know them well enough.
    Anyhow,

    Find the solutions to the following:
     sec^2x = 2secx

    I got this far...or maybe it's all wrong, I duno.
     1+tan^2x = 2secx * from a Pythagorean identity
     1+tan^2x = \frac{1}{2cosx} * from the reciprocal of sec?
     1+tan^2x (cos2x-sin2x) *cross multiplied and double angle formula? I feel this wasn't a good move, haha.

    Thanks in advanced!
    I would solve it as a quadratic rather than using an identity:

    sec^2(x) - 2sec(x) = 0

    sec(x)(sec(x)-2) = 0

    sec(x) = 0 or sec(x) = 2

    Can you finish from there?
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  3. #3
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    Quote Originally Posted by e^(i*pi) View Post
    I would solve it as a quadratic rather than using an identity:

    sec^2(x) - 2sec(x) = 0

    sec(x)(sec(x)-2) = 0

    sec(x) = 0 or sec(x) = 2

    Can you finish from there?
    Oh wow, that's much simpler.
    Further questions...
    So sec(x) = 0....1/cosx = 0...when cosine is zero, it's between pi/2 and 3pi/2, right?
    So is x=1? I think I'm a bit confused here.

    sec(x)=2
    1/cosx = 2
    No solution for this, right? Outside of unit circle?

    Hopefully I'm not too far off...:\
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  4. #4
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    e^(i*pi)'s Avatar
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    Quote Originally Posted by tiar View Post
    Oh wow, that's much simpler.
    Further questions...
    So sec(x) = 0....1/cosx = 0...when cosine is zero, it's between pi/2 and 3pi/2, right?
    So is x=1? I think I'm a bit confused here.

    sec(x)=2
    1/cosx = 2
    No solution for this, right? Outside of unit circle?

    Hopefully I'm not too far off...:\
    That sounds right, there is no determined solution for sec(x) = 0

    For sec(x) = 2 there is a solution: you can take the reciprocal of both sides:

    \frac{1}{cos(x)} = 2

    \frac{1}{\frac{1}{cos(x)}} = \frac{1}{2}

    cos(x) = 0.5

    x = \frac{\pi}{3} \text { and } \frac{4\pi}{3}

    those are the two solutions between 0 and 2pi but in general it will be

    x = \frac{\pi}{3} \pm k\pi where k is an integer
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