Sketch the Graph of h(x) = cos²2x
Question:
Sketch the graph of the following trigonometric function :
\[ h(x)=\cos^22x \]
Solution:
We know that
\[ h(x)=\cos^22x=(\cos2x)^2 \]
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
\[ \cos^22x=\frac{1+\cos4x}{2} \]
Hence the period of the graph is
\[ \frac{2\pi}{4}=\frac{\pi}{2} \]
Important properties:
- Maximum value \(=1\)
- Minimum value \(=0\)
- Period \(=\dfrac{\pi}{2}\)
- Range \(0 \le y \le 1\)
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=\cos^20=1\\[8pt] x=\frac{\pi}{8} &\Rightarrow y=\cos^2\frac{\pi}{4} =\frac12\\[8pt] x=\frac{\pi}{4} &\Rightarrow y=\cos^2\frac{\pi}{2}=0\\[8pt] x=\frac{3\pi}{8} &\Rightarrow y=\cos^2\frac{3\pi}{4} =\frac12\\[8pt] x=\frac{\pi}{2} &\Rightarrow y=\cos^2\pi=1 \end{aligned} \]
The pattern repeats after every interval
\[ \frac{\pi}{2} \]
Plot these points and draw a smooth curve through them.
Hence, the required graph is shown above.