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