A submerged scuba diver looks up toward the calm surface of a freshwater lake a d notes that the sun appears 39 degrees from the vertical. The diver’s friend is standing on the shore of the lake. At what angle above the horizon does the friend see the sun 

To solve this problem, we need to understand how light refracts as it moves from one medium to another, in this case, from water to air. This refraction causes the sun to appear at different angles for the diver underwater and the person on the shore.

### Step 1: Understanding refraction and Snell's Law

The angle of refraction when light passes from one medium to another is given by **Snell's Law**:

\[
n_1 \sin(\theta_1) = n_2 \sin(\theta_2)
\]

Where:
- \(n_1\) is the refractive index of the first medium (water in this case, approximately \(n_1 = 1.33\)),
- \(n_2\) is the refractive index of the second medium (air, approximately \(n_2 = 1.00\)),
- \(\theta_1\) is the angle of incidence (the angle from the vertical that the diver sees the sun, which is 39°),
- \(\theta_2\) is the angle of refraction (the angle that the friend on the shore sees the sun above the horizon).

### Step 2: Applying Snell's Law

We know that the angle the diver sees, \(\theta_1 = 39^\circ\), is measured from the vertical. However, Snell's Law works with the angle from the normal, so we need to convert this angle:

\[
\theta_1' = 90^\circ - 39^\circ = 51^\circ
\]

Now, using Snell's Law, we can find the angle \(\theta_2'\) from the normal in the air (which the person on the shore sees):

\[
n_1 \sin(\theta_1') = n_2 \sin(\theta_2')
\]

Substitute the known values:

\[
1.33 \sin(51^\circ) = 1.00 \sin(\theta_2')
\]

First, calculate \(\sin(51^\circ)\):

\[
\sin(51^\circ) \approx 0.777
\]

Then:

\[
1.33 \times 0.777 = \sin(\theta_2')
\]

\[
1.034 \approx \sin(\theta_2')
\]

Since \(\sin(\theta_2')\) cannot exceed 1, there's an issue with this calculation, suggesting an error with assumptions or the angles involved. Let's check

A submerged scuba diver looks up toward the calm surface of a freshwater lake and notes that the sun appears 39 degrees from the vertical. The diver’s friend is standing on the shore of the lake. At what angle above the horizon does the friend see the sun?

Learn how to calculate the angle at which a scuba diver and a friend on the shore see the sun through the phenomenon of light refraction. This problem applies Snell's Law to determine the angle of refraction from water to air. Suitable for high school students studying physics and trigonometry.

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