Find the angle of elevation of a rainbow

Question

The angle of elevation θ = 4β - 2a, where sin a = k sin β and a = 59.4 degrees and k = 1.33 is the index of refraction of water. Use the given information to find the angle of elevation θ of a rainbow. I just really don't know how to proceed. The answer should be 42 degrees

Answer 1

Hi Billy, 

 

Here's how to proceed. I'm going to add the mental process as well as the actual solution.

 

To find θ = 4β - 2a, we need to know a and β.

 

We have a = 59.4 degrees, so we just need to find β.

 

So, we use the equation sin a = k sin β to find β. Solve this equation for β by diving both sides of the equation by k to get 

 

sin β = (sin a)/k. Plug in a = 59.4 degrees and k = 1.33 and we get sin β = (sin 59.4)/1.33 = 0.6471744564

 

So, sin β = 0.6471744564.

 

To find β, we compute arcsin(0.6471755564) = 40.3 degrees (to the nearest 10th)

 

Now, substitute a = 59.4 and β = 40.3 into θ = 4β - 2a.

 

When we do this, we get θ = 4*40.3 - 2*59.4 = 42.4 degrees to the nearest 10th.

 

To the nearest degree, this is 42 degrees as you said the answer was to be.

 

I hope this helps!