sketch the graph of cosine for $\displaystyle 0<\theta<2\pi$
$\displaystyle y=2-\mid cos\frac{\theta}{2}\mid$
hi
first off , do you know how to sketch the graph of cos 1/2 Θ ?
Note that its period is 2π/(1/2)=4π
The usual graph we draw (cos Θ) has a period of 2π but this time , 4π .
|cos 1/2 Θ| - reflect the negative part of the graph to the postive side
- |cos 1/2 Θ| - reflect the whole graph about the x-axis
2 - |cos 1/2 Θ| - shift the graph up by 2 units .
$\displaystyle cos(0)=1$
$\displaystyle cos\left(\frac{{\pi}}{2}\right)=0$
$\displaystyle cos\left(\frac{\theta}{2}\right)=0\ \Rightarrow\ \frac{\theta}{2}=\frac{{\pi}}{2}\ \Rightarrow\ \theta={\pi}$
$\displaystyle cos\left(\frac{\theta}{2}\right)$ has twice the period that $\displaystyle cos\theta$ has
or it has half the frequency.
Knowing that, the graph can be drawn using $\displaystyle cos\theta$ to guide.