Problem 82 - Precalculus: Functions and Graphs

Question

The altitude of the sun is the angle \(\phi\) that the sun's rays make with the horizon at a given time and place. Determining \(\phi\) is important in tilting a solar collector to obtain maximum efficiency. On June 21 at a latitude of \(51.7^{\circ} \mathrm{N}\), the altitude of the sun can be approximated using the formula \(\sin \phi=\sin 23.5^{\circ} \sin 51.7^{\circ}+\cos 23.5^{\circ} \cos 51.7^{\circ} \cos H\) where \(H\) is called the hour angle, with \(H=-\pi / 2\) at 6 A.M., \(H=0\) at noon, and \(H=\pi / 2\) at 6 P.M. Solve the formula for \(\phi,\) and graph the resulting equation for \(-\pi / 2 \leq H \leq \pi / 2\) (b) Estimate the times when \(\phi=45^{\circ}\).

Step-by-Step Solution

Answer

Approximate times for \( \phi = 45^{\circ} \) are mid-morning and mid-afternoon.

Step 1: Understand the given formula

The formula given is \( \sin \phi = \sin 23.5^{\circ} \sin 51.7^{\circ} + \cos 23.5^{\circ} \cos 51.7^{\circ} \cos H \). It describes how the sine of the sun's altitude \( \phi \) depends on the hour angle \( H \) and two constant angles, 23.5° related to Earth's axial tilt and 51.7° related to the latitude.

Step 2: Solve for \( \phi \)

To find \( \phi \), take the inverse sine of both sides of the given equation:\[\phi = \sin^{-1} \left( \sin 23.5^{\circ} \sin 51.7^{\circ} + \cos 23.5^{\circ} \cos 51.7^{\circ} \cos H \right)\]This expression allows us to compute \( \phi \) for any value of \( H \) within the range \(-\pi/2 \leq H \leq \pi/2\).

Step 3: Convert angles to radians

Since our angle calculations and plots are in terms of \( H \) given in radians, we must convert degrees to radians for calculation consistency. Use \( 1^{\circ} = \frac{\pi}{180} \) radians:* 23.5° becomes \( \frac{23.5 \pi}{180} \)* 51.7° becomes \( \frac{51.7 \pi}{180} \)

Step 4: Graph \( \phi \) against \( H \)

Using the expression for \( \phi \), plot \( \phi = \sin^{-1} (\sin 23.5^{\circ} \sin 51.7^{\circ} + \cos 23.5^{\circ} \cos 51.7^{\circ} \cos H) \) against \( H \) from \(-\pi / 2\) to \(\pi / 2\). This will show the sun's altitude throughout the day.

Step 5: Estimate times when \( \phi = 45^{\circ} \)

To find when \( \phi = 45^{\circ} \), solve the equation \[\sin 45^{\circ} = \sin 23.5^{\circ} \sin 51.7^{\circ} + \cos 23.5^{\circ} \cos 51.7^{\circ} \cos H\]Convert 45° to radians: \( \frac{45\pi}{180} \). Determine \( H \) using an iterative method (like a graph or numerical solver) where the sun altitude \( \phi \) equals 45° for values of \( H \) within the range \(-\pi/2 \) to \(\pi/2\). Through graph inspection or solving, find approximate \( H \) values when \( \phi = 45^{\circ} \) and convert these hour angles to time.

Key Concepts

Trigonometric Functions

Trigonometric functions are the cornerstone of many mathematical concepts, including calculating the sun's altitude. In the given problem, the formula involves the sine and cosine functions to express how the altitude of the sun, denoted as \(\phi\), changes throughout the day.

Specifically, this exercise uses the following trigonometric functions:

• \(\sin\): Calculates the ratio of the opposite side to the hypotenuse in a right-angled triangle, used here to denote the sun's altitude.
• \(\cos\): Calculates the ratio of the adjacent side to the hypotenuse, signifying the variation in the sun's position relative to Earth’s tilt and latitude.

These functions help us express complex angular relationships in a more manageable way. By understanding how sine and cosine are applied, we can interpret shifts in solar altitude due to changes in time, hour angle, and geographical positioning. This understanding is critical for fields like astronomy and solar energy, where predicting solar positions is essential for optimizing efficiency.

Solar Energy

Solar energy relies heavily on accurately predicting the sun's path, which is why calculating the sun's altitude is crucial. By determining \(\phi\), the angle formed by the sun’s rays with the horizon, we can adjust solar panels to capture the most sunlight efficiently.

The axial tilt of the Earth (23.5°) and the latitude of the location (51.7°) are integral factors influencing solar energy capture. For instance, on June 21 at a latitude of 51.7°N, the sun's path directly impacts solar panel positioning. This date, marking the summer solstice in the Northern Hemisphere, has the highest solar altitude, thereby presenting opportunities for maximum energy absorption.

Properly aligning solar collectors based on this altitude ensures that they are perpendicular to the sun's rays, optimizing their efficiency. Engineers and scientists utilize equations like the one provided in the problem to make precise calculations for both residential and commercial solar energy systems.

Radians to Degrees Conversion

Understanding angle measurements in both radians and degrees is key in trigonometry and solar calculations. In this problem, the hour angle \(H\) is expressed in radians, while other essential angles are given in degrees. Converting between these two units allows for more consistent calculations.

A simple conversion formula exists:

• 1 degree = \( \frac{\pi}{180} \) radians
• Conversely, 1 radian = \( \frac{180}{\pi} \) degrees

For example, angles of 23.5° and 51.7°, related to Earth's axial tilt and latitude respectively, are converted to radians for trigonometric calculations. Without consistent units, correctly plotting or deriving outcomes from trigonometric functions like \(\sin\) and \(\cos\) would be challenging. Utilizing radians typically simplifies mathematical analysis, especially when systems or plotting software default to radians.

Hour Angle

The hour angle \(H\) is a measure of time expressed in angular terms. It plays a pivotal role in solar altitude calculations. It represents the Earth's rotation relative to the sun's position in the sky.

In the exercise, \(H\) is associated with specific times of day:

• \(H = -\frac{\pi}{2}\) radians (6 A.M.)
• \(H = 0\) radians (Noon)
• \(H = \frac{\pi}{2}\) radians (6 P.M.)

This creates a direct relationship between the sun's position and the local time, aiding in calculating its altitude at different hours. By solving the equation for \(\phi\), we trace how the altitude peaks around noon (when \(H = 0\)) and gradually decreases throughout the day, impacting what we perceive as the sun's rising and setting. Understanding the correlation between hour angle and sun altitude can assist in predicting daylight hours and optimizing solar applications.