Sketch the Graph of Φ(x) = 2 cos(x − π/6)
Question:
Sketch the graph of the following trigonometric function :
\[ \Phi(x)=2\cos\left(x-\frac{\pi}{6}\right) \]
Solution:
We know that
\[ y=\cos x \]
is the standard cosine curve.
The graph of
\[ y=2\cos\left(x-\frac{\pi}{6}\right) \]
is obtained by:
- Stretching the cosine curve vertically by factor \(2\)
- Shifting it to the right by \(\dfrac{\pi}{6}\)
Important properties:
- Amplitude \(=2\)
- Period \(=2\pi\)
- Phase shift \(=\dfrac{\pi}{6}\) to the right
- Range \(-2 \le y \le 2\)
Now calculate some important points:
\[ \begin{aligned} x=0 &\Rightarrow y=2\cos\left(-\frac{\pi}{6}\right) =\sqrt3\\[8pt] x=\frac{\pi}{6} &\Rightarrow y=2\cos0=2\\[8pt] x=\frac{2\pi}{3} &\Rightarrow y=2\cos\frac{\pi}{2}=0\\[8pt] x=\frac{7\pi}{6} &\Rightarrow y=2\cos\pi=-2\\[8pt] x=\frac{5\pi}{3} &\Rightarrow y=2\cos\frac{3\pi}{2}=0\\[8pt] x=\frac{13\pi}{6} &\Rightarrow y=2\cos2\pi=2 \end{aligned} \]
Thus the curve passes through the points
\[ (0,\sqrt3),\quad \left(\frac{\pi}{6},2\right),\quad \left(\frac{2\pi}{3},0\right),\quad \left(\frac{7\pi}{6},-2\right),\quad \left(\frac{5\pi}{3},0\right) \]
Plot these points and draw a smooth cosine curve through them.
Hence, the required graph is shown above.