No calculator: Final Destination

sin(x)*cos(x)=c, where c is a constant between 0 and 1.

Without a calculator, how do I do this?  Until now, I just graphed that shit.  But now I'm not allowed to, so I come to you for help.

sin(2x) = 2c
x = arcsin(2c)/2

Thanks, but there should be two answers.  How do I find the second one again?

I am not sure about this but usually the 'other' solution for arcsine is "pi - arcsin(...etc)"

So in this case, x = arcsin(2c)/2 and x = pi/2 - arcsin(2c)/2

Gimmie a sec to verify.

>>4 here, yeah- that check out as the answer.

Curiously enough, even though you stated that 'c' is between 0 and 1, it really must be between 0 and 0.5, because anything greater will cause a > 1 in the argument of the arcsine when evaluated, which is out of domain.

Since c is positive, both cos(x) and sin(x) should be positive, or they should both be negative. sin(x) and cos(x) are both positive for 0 < x < pi/2, and they are both negative for pi < x < 3Pi/2.

sin(x)*cos(x) = c, with 0 < c < 1 for 0 + k*pi < x < pi/2 + k*pi with k as an element of Z.

Since you are given [math]\sin x \cos x = c[\math], this implies
[eqn]\int\sin x \cos x dx = \int c dx [\eqn]
[eqn]\frac{1}{2}\sin^2 x = cx [\eqn]
[eqn] c = \frac{\sin^2 x}{2x}[\eqn].