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Math Help - trig find value

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    trig find value

    Find the exact value of tan x (without finding x) if sin x - cos x = 0.2 and sin x > 0.
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    Quote Originally Posted by Mr_Green View Post
    Find the exact value of tan x (without finding x) if sin x - cos x = 0.2 and sin x > 0.
    sin(x) - cos(x) = 0.2 <-- Divide both sides by cos(x). (So we can't let x = (pi)/2 or 3(pi)/2, for which cases tan(x) is undefined anyway.)

    tan(x) - 1 = 0.2/cos(x) = 0.2*sec(x)

    Now square both sides:
    tan^2(x) - 2*tan(x) + 1 = 0.04*sec^2(x)

    Now, tan^2(x) + 1 = sec^2(x), so:

    tan^2(x) - 2*tan(x) + 1 = 0.4*tan^2(x) + 0.4

    0.96*tan^2(x) - 2*tan(x) + 0.96 = 0

    Now use the quadratic formula:

    tan(x) = [-(-2) (+/-) sqrt{(-2)^2 - 4*0.96*0.96}]/(2*0.96)

    tan(x) = [2 (+/-) sqrt{0.3136}]/1.92

    tan(x) = [2 (+/-) 0.56]/1.92

    tan(x) = 4/3 or tan(x) = 3/4

    Which solution (or both) is it? For this we need to go back to the original equation, so we have to break the rules a little.
    tan(x) = 4/3 ==> x = 0.927295 rad
    tan(x) = 1/3 ==> x = 0.643501 rad

    By checking each of these x values in the original equation I get that only the tan(x) = 4/3 solution is acceptible.

    -Dan
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