# How do you graph cos^2 x and sin^2 x?

Oct 31, 2015

See below

#### Explanation:

Of course, without some graphic tool, you can't make an exact graph, so I'll tell you the ideas which allow you, knowing the graphs of $\cos \left(x\right)$ and $\sin \left(x\right)$, to sketch those of their squares.

• First of all, a square is always non-negative, so the graph will never go below the $x$-axis;
• Since both sine and cosine functions are bounded in $\left[- 1 , 1\right]$, their squares will be bounded in $\left[0 , 1\right]$; in fact...
• When the function is negative, it becomes positive, because you're squaring it;
• When the function is zero, its square will still be zero;
• When the function is between zero and one, its square will be between zero and one, too;
• When the function equals one, its square will equal one, too.

So, if you start from the graph of $\cos \left(x\right)$, for example, you know that ${\cos}^{2} \left(x\right)$ will have the same zeroes and the same maxima. Also, all the minima becomes maxima, because ${\left(- 1\right)}^{2} = 1$.

This is everything you can calculate perfectly. Once you have these break points, you must connect them with a line that resembles the one of $\cos \left(x\right)$, and you can't to anything more precise