Question

Angle of minimum deviation for a prism of refractive index 1.5, is equal to the angle of the prism. Then the angle of the prism is: $(\text{given co}{\text{s}}^{0}41=0.75)$

Answer 1

Hint: In this question use the direct relationship between the angle of prism (A), angle of minimum deviation (${\theta }_m$) and refractive index of the prism, that is ${\mu }_p={\displaystyle \frac{\sin \left({\displaystyle \frac{A+{\theta }_m}{2}}\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}}$. Then use the constraints given in the question to find the value of the angle of the prism.

Complete step-by-step answer:
Let the angle of minimum deviation for a prism be ${\theta }_m$.
Let the angle of the prism = A.

Now as we know the relation of refractive index of the prism, angle of minimum deviation of the prism and angle of the prism is given as

$\Rightarrow {\mu }_p={\displaystyle \frac{\sin \left({\displaystyle \frac{A+{\theta }_m}{2}}\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}}$, where ${\mu }_p$ is the refractive index of the prism.
Now it is given that the angle of minimum deviation of the prism is equal to the angle of the prism.

$\Rightarrow {\theta }_m=A$ And ${\mu }_p=1.5$
So substitute these values in the given equation we have,
$\Rightarrow 1.5={\displaystyle \frac{\sin \left({\displaystyle \frac{A+A}{2}}\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}}$
$\Rightarrow 1.5={\displaystyle \frac{\sin \left(A\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}}$
Now as we know that $\sin 2x=2\sin x\cos x$ so use this property in above equation we have,
$\Rightarrow 1.5={\displaystyle \frac{2\sin \left({\displaystyle \frac{A}{2}}\right)\cos \left({\displaystyle \frac{A}{2}}\right)}{\sin \left({\displaystyle \frac{A}{2}}\right)}}=2\cos \left({\displaystyle \frac{A}{2}}\right)$
$\Rightarrow \cos {\displaystyle \frac{A}{2}}={\displaystyle \frac{1.5}{2}}=0.75$
$\Rightarrow {\displaystyle \frac{A}{2}}={\cos }^{-1}\left(0.75\right)$
Now it is given that 0.75 = $\cos {41}^o$ so use this value we have,
$\Rightarrow {\displaystyle \frac{A}{2}}={\cos }^{-1}\left(\cos {41}^o\right)={41}^o$
$\Rightarrow A=2\times {41}^o={82}^o$

So this is the required angle of the prism.
Hence option (C) is the correct answer.

Note: Sometimes the knowledge of trigonometric identities also helps in simplification of the problems of this kind, so it is advised to have a good gist of the trigonometric identities some of them are being mentioned above like $\sin 2x=2\sin x\cos x$. The other includes $\cos 2x={\cos }^{2}x-{\sin }^{2}x$, ${\sin }^{2}x=1-{\cos }^{2}x$, $1+{\tan }^{2}x={\sec }^{2}x$ etc.