Ch. 2 - Acute Angles and Right Triangles

Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook

Lial 12th Edition

Ch. 2 - Acute Angles and Right Triangles

Problem 2.3.83

Chapter 3, Problem 2.3.83

(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.

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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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Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.

cos θ = 0.85536428

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CONCEPT PREVIEW Match each trigonometric function value or angle in Column I with its appropriate approximation in Column II.

Column I: 1.

cot⁻¹ 30

Column II:

A. 88.09084757°

B. 63.25631605°

C. 1.909152433°

D. 17.45760312°

E. 0.2867453858

F. 1.962610506

G. 14.47751219°

H. 1.015426612

I. 1.051462224

J. 0.9925461516

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