Sketch the Graph of g(x) = cos 2πx
Question:
Sketch the graph of the following trigonometric function :
\[ g(x)=\cos 2\pi x \]
Solution:
We know that
\[ y=\cos x \]
is the standard cosine curve.
In the function
\[ y=\cos 2\pi x \]
the angle is multiplied by \(2\pi\). Therefore the graph oscillates more rapidly.
Important properties:
- Amplitude \(=1\)
- Period \(=\dfrac{2\pi}{2\pi}=1\)
- Range \(-1 \le y \le 1\)
Thus one complete cosine wave is obtained in the interval
\[ 0 \le x \le 1 \]
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=\cos0=1\\[8pt] x=\frac14 &\Rightarrow y=\cos\frac{\pi}{2}=0\\[8pt] x=\frac12 &\Rightarrow y=\cos\pi=-1\\[8pt] x=\frac34 &\Rightarrow y=\cos\frac{3\pi}{2}=0\\[8pt] x=1 &\Rightarrow y=\cos2\pi=1 \end{aligned} \]
Thus the curve passes through the points
\[ (0,1),\quad \left(\frac14,0\right),\quad \left(\frac12,-1\right),\quad \left(\frac34,0\right),\quad (1,1) \]
Plot these points and draw a smooth cosine curve through them.
Hence, the required graph is shown above.