# Chapter 22 - Problems - Page 865: 51

We can see the graph of $I = I_0~cos^2~\theta$ below. We can use Malus' law to sketch the graph of $I$ as a function of $\theta$. Note that $I = I_0~cos^2~\theta$ We can find $I$ for these angles $\theta$: When $\theta = 0^{\circ}$, $I = I_0~cos^2~0^{\circ} = I_0$ When $\theta = 30^{\circ}$, $I = I_0~cos^2~30^{\circ} = \frac{3I_0}{4}$ When $\theta = 45^{\circ}$, $I = I_0~cos^2~45^{\circ} = \frac{I_0}{2}$ When $\theta = 60^{\circ}$, $I = I_0~cos^2~60^{\circ} = \frac{I_0}{4}$ When $\theta = 90^{\circ}$, $I = I_0~cos^2~90^{\circ} = 0$ When $\theta = 120^{\circ}$, $I = I_0~cos^2~120^{\circ} = \frac{I_0}{4}$ When $\theta = 135^{\circ}$, $I = I_0~cos^2~135^{\circ} = \frac{I_0}{2}$ When $\theta = 150^{\circ}$, $I = I_0~cos^2~150^{\circ} = \frac{3I_0}{4}$ When $\theta = 180^{\circ}$, $I = I_0~cos^2~180^{\circ} = I_0$ When $\theta = 210^{\circ}$, $I = I_0~cos^2~210^{\circ} = \frac{3I_0}{4}$ When $\theta = 225^{\circ}$, $I = I_0~cos^2~225^{\circ} = \frac{I_0}{2}$ When $\theta = 240^{\circ}$, $I = I_0~cos^2~240^{\circ} = \frac{I_0}{4}$ When $\theta = 270^{\circ}$, $I = I_0~cos^2~270^{\circ} = 0$ When $\theta = 300^{\circ}$, $I = I_0~cos^2~300^{\circ} = \frac{I_0}{4}$ When $\theta = 315^{\circ}$, $I = I_0~cos^2~315^{\circ} = \frac{I_0}{2}$ When $\theta = 330^{\circ}$, $I = I_0~cos^2~330^{\circ} = \frac{3I_0}{4}$ When $\theta = 360^{\circ}$, $I = I_0~cos^2~360^{\circ} = I_0$ We can see the graph of $I = I_0~cos^2~\theta$ below. 