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

1 Answer
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(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