Question

For sin x > 0, \(2\sin^{2}(x) + \sin(x)\cos(x) = 0\). Give the value of |\(\cos(x)\)|. Please show all work!

Answer 1

\[
2\sin^2(x)+\sin(x)\cos(x)=0\\
\sin(x)[2\sin(x)+\cos(x)]=0
\]
since it is known that \(\sin(x)>0\), we have \(\sin(x)\ne0\)

Answer 2

hence,
\[2\sin(x)+\cos(x)=0\\
\cos(x)=-2\sin(x)
\]

Answer 3

From there, is it \(\tan(x) = -\frac{1}{2}\)?

Answer 4

now, we know that
\[\sin^2(x)+\cos^2(x)=1\\
{1\over4}\cos^2(x)+\cos^2(x)=1\\
\cos^2(x)={4\over5}\\
\Large\boxed{|\cos(x)|=\sqrt{4\over5}}
\]

Answer 5

Oh, I get it now. I didn't realize that part with tan x. Thank you!

Answer 6

no we are concerned only with obtaining the "cos(x)"
so, we exhaust ourselves with the above "sin(x)" and "cos(x)" relation with another "sin(x)" and "cos(x)" relation and eliminate the "sin(x)" and we are left with "cos(x)"
we should not bring the unnecesdsary ratios in there!! :)