Using a double angle formula with Snell's Law

smeiste

1. The problem statement, all variables and given/known data
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.


2. Relevant 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

3. 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

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

ehild

smeiste

but then how do you solve for a?

ehild

Show your equation.

ehild

smeiste

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

gneill

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

bobquantum

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.

smeiste

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

gneill

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

smeiste

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

gneill

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?

smeiste

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

smeiste

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

smeiste

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!