Question

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

Answer 1

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(x)` and `sin(x)`, 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 `[-1,1]`, their squares will be bounded in `[0,1]`; 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(x)`, for example, you know that `cos^2(x)` will have the same zeroes and the same maxima. Also, all the minima becomes maxima, because `(-1)^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(x)`, and you can't to anything more precise