Refraction and Lenses Legacy Problem #3 Guided Solution

One of the aspects of physics that I'm particularly fond of is its predictive ability, its ability to predict an outcome if given some initial known parameters. The refraction of light at a boundary is an example of a case in which physics is able to predict some information. It just seems that nature follows very predictable laws, and the law which governs the refraction of light at the boundary is known as Snell's law. It goes like this in equation form, n1 times the sine of theta 1 equals n2 times the sine of theta 2. And in the equation, the n values are the indexes of refraction, or the index of refraction, of the given materials on each side of the boundary. Here in this problem, 1.00 for air and 1.33 for water. And the theta values are the angles that the light rays make with the normal line. The angle of incidence and the angle of refraction, or the angle in material 1, theta 1, and the angle in material 2, theta 2. And the sine is a function from trigonometry that we use, that we apply to angles. So if we know three of the four quantities, we can predict the fourth quantity. And here in this question, we know the n of air is 1.00, and the angle of air is 45 degrees. You can call that n1 and theta 1. And we know the n in water, which we can call n2. And we're looking for the angle of refraction in the water, which we can call theta 2. So now all you need to do is plug those three numbers into the Snell's Law equation and calculate the fourth. Doing so begins with 1.00 times the sine of 45 degrees equals 1.33 times the sine of theta r, where theta r, theta 2, is my unknown value. Now I can evaluate, I can divide both sides by 1.33 and evaluate the left side of the equation. It comes out 1.00 times sine of 45 degrees divided by 1.33, comes out to be 0.53166. There is some angle for which the sine value is 0.53166, and if we wish to determine that angle value that gives you that sine value, then what you need to do is apply the inverse sine function to 0.53166. The inverse sine function is often written on calculators as sin-1. You'll often find it just above the sine button on your calculator. Look there now. To employ it, you simply go second sin. That's on your many-year scientific calculators and most of your TI graphing calculators. That's all you have to do. Second sin. If you're using a graphing calculator, you go second sin and then you enter the number 0.53166, and it will give you 32.1176 as your degree measure. If you're using a scientific calculator, you simply have to have the 0.53166 in the calculator first and then you press second sin and you're typically able to get the same answer, 32.1176. I can round this to three significant digits such that the answer is 32.1 degrees. Now if you're not getting this, the first place to check is the angle mode of your calculator. On your graphing calculators, there's a mode button. You hit mode and then you arrow down where you see radians and degrees. You want to make sure that degrees is selected, because this 45 is in units of degrees. And then on your scientific calculator, there's usually a DRG button of some sort. And the DRG button allows you to toggle between degrees and radians and gradients. So you want to, again, make sure it's in degrees.