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A narrow beam of light is incident from air onto a glass surface with index of refraction 1.56. Find the angle of incidence for which the corresponding angle of refraction is one-half the angle of incidence. Hint: You might want to use the trigonometric identity $\sin 2 \theta=2 \sin \theta \cos \theta$.

$77.5^{\circ}$

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So for our question were told that we have a ray of light going from air to glass, where the index of refraction of glass, which I call in sub D, is 1.56 It also tells us that the angle of refraction is half the angle of incident or fatal. One is equal to two beta two. Okay, So using Snell's law, we can, ah then find the angle of incident and gives us the hint of using that triggered a metric identity there, which we are gonna make use of. So Snell's law here says in one times the sign of the incident angle, which we're gonna call tooth data. So this this data term is what we're gonna find here is equal to in our situation into timesthe Sign of data. Okay, so here in one is air, so we could replace that with instead, Eh? Times the sign of tooth ada is equal to in sub G. That's our second medium. Under the sign of data. Well, that triggered a metric identity. Says that sign to fade A is equal to two times that co sign of feta multiplied by the sign of data and I'm gonna divide both sides by the index of refraction of air, which is equal to one. So anything divided by one is just itself. So we can just get rid of the index of refraction of air. So this is gonna be equal to insert g times the sign of data. Okay, well, sign of fate is on both sides of the equation, so we can just cancel that out and we can go ahead and solve for theta, which is going to be equal to the inverse co. Sign co signed an A minus one of 1/2 the index of refraction of the glass. So in sub d, divided by two, which we find data here, then is equal to set, um, 38.74 degrees. But we were told that the angle of incidence is two times data, so we just found fatal, which was equal to fade A too. So we need to find two times data. We ran out of room here so we can say fatal one is equal to two times data, which is equal to 77.5 degrees. And that is the solution to our question.