Sketch the Graphs of y = cos²x and y = cos x on the Same Axes
Question:
Sketch the graphs of the following curves on the same scale and the same axes :
\[ y=\cos^2x \]
\[ y=\cos x \]
Solution:
We know that
\[ y=\cos x \]
is the standard cosine curve having amplitude \(1\) and period \(2\pi\).
Now consider
\[ y=\cos^2x \]
Since square of cosine is always non-negative, the graph always lies above the x-axis.
Using the identity
\[ \cos^2x=\frac{1+\cos2x}{2} \]
the period of \[ y=\cos^2x \] is
\[ \pi \]
Hence the graph of \[ y=\cos^2x \] completes two waves in the interval \[ 0 \le x \le 2\pi \]
Important points for \[ y=\cos x \] are:
\[ (0,1),\quad \left(\frac{\pi}{2},0\right),\quad (\pi,-1),\quad \left(\frac{3\pi}{2},0\right),\quad (2\pi,1) \]
Important points for \[ y=\cos^2x \] are:
\[ (0,1),\quad \left(\frac{\pi}{4},\frac12\right),\quad \left(\frac{\pi}{2},0\right),\quad \left(\frac{3\pi}{4},\frac12\right),\quad (\pi,1) \]
and the pattern repeats up to \(2\pi\).
Plot these points and draw smooth curves on the same coordinate axes.
Hence, the required graphs are shown above.
Graph Features:
- Amplitude of \(y=\cos x\) is \(1\)
- Range of \(y=\cos^2x\) is \(0 \le y \le 1\)
- Period of \(y=\cos x\) is \(2\pi\)
- Period of \(y=\cos^2x\) is \(\pi\)
- \(y=\cos^2x\) always lies above the x-axis