Chapter 10.7, Problem 3MRPS

Solution

a.

To find: The simplified form of given formula.

The simplified form is $n=\cos \frac{\alpha }{2}+\sin \frac{\alpha }{2}\cot \frac{\theta }{2}$ .

Given:

  $n=\frac{\sin \left(\frac{\theta }{2}+\frac{\alpha }{2}\right)}{\sin \frac{\theta }{2}}$

Concept used:

The sum formula:

  $\sin \left(x+y\right)=\sin x\cos y+\cos x\sin y$ .

Calculation:

Consider,

  $n=\frac{\sin \left(\frac{\theta }{2}+\frac{\alpha }{2}\right)}{\sin \frac{\theta }{2}}$

By using the sum formula, it can be written as

  $\begin{array}{c}n=\frac{\sin \frac{\theta }{2}\cos \frac{\alpha }{2}+\cos \frac{\theta }{2}\sin \frac{\alpha }{2}}{\sin \frac{\theta }{2}}\\ n=\frac{\sin \frac{\theta }{2}\cos \frac{\alpha }{2}}{\sin \frac{\theta }{2}}+\frac{\cos \frac{\theta }{2}\sin \frac{\alpha }{2}}{\sin \frac{\theta }{2}}\\ n=\cos \frac{\alpha }{2}+\sin \frac{\alpha }{2}\cdot \frac{\cos \frac{\theta }{2}}{\sin \frac{\theta }{2}}\\ n=\cos \frac{\alpha }{2}+\sin \frac{\alpha }{2}\cot \frac{\theta }{2}\end{array}$

Hence, the simplified form is $n=\cos \frac{\alpha }{2}+\sin \frac{\alpha }{2}\cot \frac{\theta }{2}$ .

b.

To find: The value of $\theta$ .

  $\theta \approx \text{76}{\text{.5}}^{\circ }$ .

Given:

  $n=\frac{\sin \left(\frac{\theta }{2}+\frac{\alpha }{2}\right)}{\sin \frac{\theta }{2}}$ , $\alpha ={60}^{\circ }$ , and the prism is made of glass.

Calculation:

The index of refraction of the glass is 1.5.

Now, by substituting $n=1.5,\alpha ={60}^{\circ }$ in the given formula, it follows that

  $\begin{array}{c}1.5=\frac{\sin \left(\frac{\theta }{2}+\frac{{60}^{\circ }}{2}\right)}{\sin \frac{\theta }{2}}\\ 1.5=\frac{\sin \left(\frac{\theta }{2}+{30}^{\circ }\right)}{\sin \frac{\theta }{2}}\end{array}$

Now by using the graphing calculator, it follows that

  $\theta \approx \text{76}{\text{.5}}^{\circ }$ .