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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- April 28th 2007, 09:31 AMMr_Greentrig find value
Find the exact value of tan x (without finding x) if sin x - cos x = 0.2 and sin x > 0.

- April 28th 2007, 09:49 AMtopsquark
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