# Using a double angle formula with Snell's Law

## Homework Statement

A light ray is incident from air onto a glass surface with an index of refraction n = 1.56. Find the angle of incidence for which the corresponding angle of refraction is one-half the angle of incidence. Both angles are defined with the normal to the surface.

## Homework Equations

n = sin a1/sin a2 (Snell's Law)

sin(a + b) = (sin a x cos b) + (cos a x sinb)

sin(2a) = 2 sin a cos a

## The Attempt at a Solution

Somehow you plug in the double angle formula into snells law. but I don't understand how this is a down. a breakdown of the equations used to solve problem would be extremely helpful :)

ehild
Homework Helper
The angle of incidence is 2a, the angle of refraction is a. Plug in to Snell's law.

ehild

but then how do you solve for a?

ehild
Homework Helper

ehild

sin(a)/sin(2a) = 1.56

gneill
Mentor
sin(a)/sin(2a) = 1.56

What happens if you apply the sine double angle formula here?

Your equation is actually incorrect. by Snell's formula,
sin a = 1.56Apply the duble angle formula to sin (2a) to get 2sin(a)cos(a). since a does not equal an integral multiple of pi, you can divide out sin(a). From there, you can find that cos(a) = 0.321
Use the inverse of cosine function to solve for a = 71.306 degrees.

Last edited:
if sin a = 1.56, how do you apply the double angle formula?

gneill
Mentor
Your equation is actually incorrect. by Snell's formula,
sin a = 1.56Apply the duble angle formula to sin (2a) to get 2sin(a)cos(a). since a does not equal an integral multiple of pi, you can divide out sin(a). From there, you can find that cos(a) = 0.321
Use the inverse of cosine function to solve for a = 71.306 degrees.

Snell's Law:
$$\frac{sin(\theta 1)}{sin(\theta 2)} = \frac{v1}{v2} = \frac{n2}{n1}$$

im so confused.. could someone just show the steps on how to solve it? i think much better in equations than words.

gneill
Mentor
Take Snell's law as I wrote it above (You can ignore the velocity ratio, I only included it for completeness. It's not needed for this problem). Substitute the given values for the angles and indexes of refraction. What do you get?

isnt that what i wrote before? sin a/sin 2a = 1.56/1.

sorry, i see. sin2a/sina = 1.56/1

ah ha! so sin 2a = 2sinacosa and the sins cancel out, giving you a final equation of cos a = 0.78. thanks very much everyone!