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Math Help - using identity to solve for x

  1. #1
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    using identity to solve for x

    the question says: Use an identity to find solutions on the interval [0,2pi]

    sec^2 x + tanx = 3

    thanks
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  2. #2
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    Hello, kenan!

    Use an identity to find solutions on the interval [0,\,2\pi]

    . . \sec^2\!x + \tan x \:= \:3
    Use: . \sec^2\!\theta \;=\;\tan^2\!\theta + 1

    Then we have: . \tan^2\!x + 1 + \tan x \;=\;3

    . . which is a quadratic:] . \tan^2\!x + \tan x - 2 \:=\:0

    . . which factors: . (\tan x -1)(\tan x + 2)\:=\:0


    Then we have:
    \tan x -1 \:=\:0\quad\Rightarrow\quad \tan x \:=\:1\quad\Rightarrow\quad x \:=\:\frac{\pi}{4},\:\frac{5\pi}{4}

    \tan x + 2 \:=\:0\quad\Rightarrow\quad \tan x \:=\:-2\quad\Rightarrow\quad x \:=\:\tan^{-1}(-2) \:\approx\:2.034,\;5.176

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  3. #3
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    hey thanks for the answer, but can you show me how you derived
    <br />
\sec^2\!\theta \;=\;\tan^2\!\theta + 1<br />

    ive never seen that identity before so I'm just curious where it comes from
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