Problem 69 - Algebra and Trigonometry

Question

Rainbows Rainbows are created when sunlight of different wavelengths (colors) is refracted and reflected in raindrops. The angle of elevation \(\theta\) of a rainbow is always the same. It can be shown that \(\theta=4 \beta-2 \alpha\) where $$\sin \alpha=k \sin \beta$$ and \(\alpha=59.4^{\circ}\) and \(k=1.33\) is the index of refraction of water. Use the given information to find the angle of elevation \(\theta\) of a rainbow. (For a mathematical explanation of rainbows see Calculus, 5 th Edition, by James Stewart, pages \(288-289 .\) )

Step-by-Step Solution

Answer

The angle of elevation \( \theta \) of a rainbow is \( 42^\circ \).

Step 1: Understand the Given Equation

We start with the given equation for the angle of elevation of a rainbow, \( \theta = 4\beta - 2\alpha \), where \( \alpha = 59.4^\circ \) and the equation \( \sin \alpha = k \sin \beta \) with \( k = 1.33 \).

Step 2: Substitute \( \alpha \) into the Equation

Given \( \alpha = 59.4^\circ \), we substitute this into the equation \( \sin \alpha = k \sin \beta \): \[ \sin 59.4^\circ = 1.33 \sin \beta \]

Step 3: Calculate \( \sin \alpha \)

Calculate the sine of \( \alpha = 59.4^\circ \): \[ \sin 59.4^\circ \approx 0.8571 \]

Step 4: Solve for \( \sin \beta \)

Using \( \sin 59.4^\circ = 1.33 \sin \beta \), solve for \( \sin \beta \): \[ \sin \beta = \frac{0.8571}{1.33} \approx 0.6446 \]

Step 5: Find \( \beta \) Using Inverse Sine

Calculate \( \beta \) by taking the inverse sine of 0.6446: \[ \beta = \arcsin(0.6446) \approx 40.2^\circ \]

Step 6: Substitute \( \alpha \) and \( \beta \) into \( \theta \) Equation

Substitute \( \alpha = 59.4^\circ \) and \( \beta = 40.2^\circ \) into the equation for \( \theta \): \[ \theta = 4(40.2) - 2(59.4) \]

Step 7: Calculate \( \theta \)

Perform the calculations: \[ \theta = 160.8 - 118.8 = 42^\circ \]

Key Concepts

Refraction

Refraction is a fascinating phenomenon that occurs when light waves pass from one medium into another, changing speed and direction in the process. This happens due to variations in the density of different materials. For instance, when light enters water from the air, it slows down and bends, resulting in refraction. Rainbows are a beautiful example of refraction, where light is both refracted and reflected within raindrops.

• Light slows down when entering denser mediums like water.
• This causes the light to bend, changing direction.
• Refraction plays a crucial role in forming rainbows.

Understanding refraction is essential when studying optical phenomena.
It involves Snell's Law, which relates the angles of incidence and refraction with the indices of refraction of the two media involved.

Angle of elevation

The angle of elevation is the angle formed between the horizontal plane and a line that points upwards to an object, in this case, the rainbow. In our rainbow exercise, the angle of elevation is calculated using the relationship \[\theta = 4\beta - 2\alpha\]Rainbows have a constant angle of elevation, which makes understanding this angle significant in various scientific studies.

• This angle helps determine how high above the horizon the rainbow appears.
• It provides insights into the path that light takes through raindrops.
• The angle is often referenced in trigonometric calculations.

Knowing the angle of elevation is crucial for predicting and identifying the appearance of natural phenomena like rainbows.

Inverse sine

The inverse sine function, written as \( \arcsin \), is a trigonometric function that helps find the angle corresponding to a given sine value. It is essential when we need to calculate angles, like in our exercise, where we found \( \beta \) using \( \arcsin(0.6446) \).
Inverse sine is significant because:

• It allows us to determine the original angle from the sine value.
• This is vital when solving trigonometric equations.
• The function is limited to values between \(-1\) and \(1\), making it bounded.

Applying inverse sine ensures accuracy in determining angles.
It's a common tool in trigonometry and geometry.

Index of refraction

The index of refraction, denoted as \( k \), measures how much light slows down when passing through a medium. It is a key factor in understanding refraction. In our exercise, the index of refraction for water is \( 1.33 \), signifying that light travels slower in water compared to air.
The index of refraction is crucial:

• It determines the bending of light as it enters a different medium.
• Higher indices indicate a greater degree of bending.
• It can be utilized in Snell's Law to predict angles of refraction.

Knowing the index of refraction helps accurately calculate light paths.
It's foundational in designing lenses and optics.