How do I calculate the refracted angle using Snell’s law?

Use Snell’s law: n₁ sin θ₁ = n₂ sin θ₂. Here, n₁ and θ₁ refer to the first medium and incoming ray; n₂ and θ₂ refer to the second medium and refracted ray. Both angles are measured from the normal—the imaginary line perpendicular to the boundary—not from the boundary itself. [S1]

Rearrange to isolate the unknown angle: θ₂ = sin⁻¹[(n₁/n₂) sin θ₁]

Substitute the indices and incident angle, calculate the expression inside the inverse sine, and then take sin⁻¹. Use a calculator in the angle unit your problem expects, commonly degrees.

Assume light travels from a medium with n₁ = 1.00 into one with n₂ = 1.50, and the incident angle is θ₁ = 30°. These are example values, not measurements of a particular setup.

θ₂ = sin⁻¹[(1.00/1.50) sin 30°]

Since sin 30° = 0.5, the inverse-sine input is (1/1.5) × 0.5 ≈ 0.3333. Thus θ₂ = sin⁻¹(0.3333) ≈ 19.5°.

Sketch (not to scale):

          Medium 1 (n₁ = 1.00)
                incoming ray
                     \
                      \  θ₁ = 30°
                       \
                        •
                        |\
             normal     | \  θ₂ ≈ 19.5°
                        |  \
------------------------+---------------- boundary
          Medium 2 (n₂ = 1.50)

The vertical normal is perpendicular to the horizontal boundary. Read both angles between the ray and that normal. The calculation gives a smaller angle in this example.

First check the setup: identify which medium the ray starts in, pair each index with its angle, and confirm that the given angle is measured from the normal. Then recalculate the ratio and sine, and verify your calculator’s angle mode. A boundary angle and an angle from the normal are complementary, so confusing the two can change the result.

The inverse-sine expression gives a transmitted angle only if its input is between −1 and 1. If it is outside that range, there is no real transmitted angle from that calculation. For light going from a higher-index medium to a lower-index one, this can indicate the critical-angle condition and total internal reflection; it is not a reason to force an angle by rounding. [S1] The supplied information alone does not establish the details of a particular physical setup.

Write the known values with the medium labels, draw the boundary and normal, and mark the incident angle from the normal. Then show the rearranged formula before substituting values. Keep a few calculator digits until the final step, and round the reported angle to a sensible precision for the given inputs. If the problem does not specify units or rounding, state the assumption you used.

For the example, a compact solution is: “Assume θ₁ = 30° is measured from the normal. With n₁ = 1.00 and n₂ = 1.50, θ₂ = sin⁻¹[(1.00/1.50)sin 30°] ≈ 19.5°.” This layout makes the angle convention, substitution, and result visible. The source supplies the relationship and angle convention; it does not establish the values for your particular problem, so use the values stated in that problem. [S1]