Sketch the Graph of u(x) = cos²(x/2)
Question:
Sketch the graph of the following trigonometric function :
\[ u(x)=\cos^2\frac{x}{2} \]
Solution:
We know that
\[ u(x)=\cos^2\frac{x}{2} =\left(\cos\frac{x}{2}\right)^2 \]
Since square of cosine is always non-negative, the graph always lies above the x-axis.
Using the identity
\[ \cos^2\theta=\frac{1+\cos2\theta}{2} \]
we get
\[ \cos^2\frac{x}{2} =\frac{1+\cos x}{2} \]
Important properties:
- Maximum value \(=1\)
- Minimum value \(=0\)
- Period \(=2\pi\)
- Range \(0 \le y \le 1\)
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=\cos^20=1\\[8pt] x=\frac{\pi}{2} &\Rightarrow y=\cos^2\frac{\pi}{4} =\frac12\\[8pt] x=\pi &\Rightarrow y=\cos^2\frac{\pi}{2}=0\\[8pt] x=\frac{3\pi}{2} &\Rightarrow y=\cos^2\frac{3\pi}{4} =\frac12\\[8pt] x=2\pi &\Rightarrow y=\cos^2\pi=1 \end{aligned} \]
Thus the curve passes through the points
\[ (0,1),\quad \left(\frac{\pi}{2},\frac12\right),\quad (\pi,0),\quad \left(\frac{3\pi}{2},\frac12\right),\quad (2\pi,1) \]
Plot these points and draw a smooth curve through them.
Hence, the required graph is shown above.