# What is the graph of y=sin^2 x?

Mar 23, 2015

Here is the graph:
graph{(sinx)^2 [-10, 10, -5, 5]}

Recall the double-angle formula for cosine:
$\cos \left(2 x\right) = 1 - 2 {\sin}^{2} \left(x\right)$

Subtract $1$ from both sides:
$\cos \left(2 x\right) - 1 = - 2 {\sin}^{2} \left(x\right)$

Divide both sides by $- 2$
$- \frac{1}{2} \cos \left(2 x\right) + \frac{1}{2}$

You now have a standard cosine equation with

Amplitude = $\frac{1}{2}$
Period = $\pi$
Vertical Shift = up by $\frac{1}{2}$

Something important to recognize, if you compare this to the graph of $y = {\cos}^{2} \left(x\right)$,(http://socratic.org/questions/what-is-the-graph-of-y-cos-2-x) you can see one is simply the negative of the other.