3. Unit Circle

Trigonometric Functions on the Unit Circle

3. Unit Circle

Trigonometric Functions on the Unit Circle

2:38 minutes

Problem 2.3.83

Textbook Question

(Modeling) Fish's View of the World The figure shows a fish's view of the world above the surface of the water. (Data from Walker, J., 'The Amateur Scientist,' Scientific American.) Suppose that a light ray comes from the horizon, enters the water, and strikes the fish's eye. Assume that this ray gives a value of 90° for angle θ₁ in the formula for Snell's law. (In a practical situation, this angle would probably be a little less than 90°.) The speed of light in water is about 2.254 x 10⁸ m per sec. Find angle θ₂ to the nearest tenth.

Verified step by step guidance

Identify the given information: the angle of incidence $ \theta_1 = 90^\circ $ (the angle the light ray makes with the normal as it enters the water), and the speed of light in water $ v_2 = 2.254 \times 10^8 $ m/s. The speed of light in air $ v_1 $ is approximately $ 3.00 \times 10^8 $ m/s.

Recall Snell's Law, which relates the angles and speeds of light in two media: \[ \\sin(\theta_1) \\times v_1 = \\sin(\theta_2) \\times v_2 \]

Since $ \theta_1 = 90^\circ $, $ \\sin(90^\circ) = 1 $, so the equation simplifies to: \[ v_1 = \\sin(\theta_2) \\times v_2 \]

Solve for $ \\sin(\theta_2) $: \[ \\sin(\theta_2) = \\frac{v_1}{v_2} \]

Calculate $ \theta_2 $ by taking the inverse sine (arcsin) of the ratio: \[ \theta_2 = \\arcsin \\left( \\frac{v_1}{v_2} \\right) \] Round the result to the nearest tenth of a degree.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Snell's Law

Snell's Law describes how light bends when it passes between two media with different refractive indices. It states that n₁ sin(θ₁) = n₂ sin(θ₂), where n₁ and n₂ are the refractive indices and θ₁ and θ₂ are the angles of incidence and refraction. This law is essential for calculating the angle of refraction when light enters water from air.

Refractive Index and Speed of Light in Media

The refractive index (n) of a medium is the ratio of the speed of light in a vacuum to its speed in that medium. Since light travels slower in water than in air, water has a higher refractive index. Knowing the speed of light in water allows calculation of its refractive index, which is crucial for applying Snell's Law.

Critical Angle and Total Internal Reflection

The critical angle is the angle of incidence above which light cannot pass through the boundary and is instead totally internally reflected. When θ₁ approaches 90°, it relates to the critical angle scenario, affecting how light rays from the horizon enter the water. Understanding this helps interpret the physical meaning of the given angle in the problem.

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Textbook Question

(Modeling) Speed of Light When a light ray travels from one medium, such as air, to another medium, such as water or glass, the speed of the light changes, and the light ray is bent, or refracted, at the boundary between the two media. (This is why objects under water appear to be in a different position from where they really are.) It can be shown in physics that these changes are related by Snell's law c₁ = sin θ₁ , c₂ sin θ₂ where c₁ is the speed of light in the first medium, c₂ is the speed of light in the second medium, and θ₁ and θ₂ are the angles shown in the figure. In Exercises 81 and 82, assume that c₁ = 3 x 10⁸ m per sec. Find the speed of light in the second medium for each of the following. a. θ₁ = 46°, θ₂ = 31° b. θ₁ = 39°, θ₂ = 28°

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Use a calculator to approximate the value of each expression. Give answers to six decimal places. tan 11.7689°

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Multiple Choice

Find the sine, cosine, and tangent of each angle using the unit circle.

$$\theta$=-1.18$ rad, $$\left$($\frac{5}{13}$,-$\frac{12}{13}\]\right$)$