### Understanding the Angle of Elevation of the SunThe angle of elevation of the sun (\(\theta\)) in the sky at any latitude (\(L\)) is calculated using the formula below. Here's how it works:\[\sin \theta = \cos D \cos L \cos \omega + \sin D \sin L\]#### Explanation of Terms:- \(\theta = 0\) corresponds to sunrise.- \(\theta = \frac{\pi}{2}\) occurs if the sun is directly overhead.- \(\omega\) is the number of radians that Earth has rotated through since noon, with \(\omega = 0\) at noon.- \(D\) is the declination of the sun, which varies because Earth is on a tilted axis.#### Example Calculation:Suppose we are asked to find the angle of elevation (\(\theta\)) of the sun at 11 P.M. on July 4, 2012, for Town A, where:- The latitude \(L = 46.30^\circ\) or \(0.8081\) radian.- At that time, \(D \approx 0.3555\) and \(\omega \approx 2.8798\).Using the formula, we substitute the given values:\[\sin \theta = \cos 0.3555 \cos 0.8081 \cos 2.8798 + \sin 0.3555 \sin 0.8081\]Evaluating this expression gives the angle \(\theta\). This calculation involves trigonometric functions and requires a calculator or computational tool for precise results.### Diagram Description:There are no graphs or diagrams in the provided text.This information aids in understanding how the angle of elevation of the sun can be calculated using specific parameters such as declination and Earth's rotation since noon.

The angle of elevation of the sun in the sky at any latitude L is calculated with the formula below. In the formula, 0=0 corresponds to sunrise and 0= occurs if the sun is directly overhead. The greek letter omega, , is the number of radians that Earth has rotated through since noon, when = 0. D is the declination of the sun, which varies because Earth is on a tilted axis. Town A has a latitude L= 46.30°, or 0.8081 radian. Find the angle elevation 0 of the sun at 11 P.M. on July 4, 2012, where at that time D≈ 0.3555 and 2.8798. sin 0= cos D cos L cos + sin D sin L

The angle of elevation of the sun in the sky at any latitude L is calculated with the formula below. In the formula, 0=0 corresponds to sunrise and 0= occurs if the sun is directly overhead. The greek letter omega, , is the number of radians that Earth has rotated through since noon, when = 0. D is the declination of the sun, which varies because Earth is on a tilted axis. Town A has a latitude L= 46.30°, or 0.8081 radian. Find the angle elevation 0 of the sun at 11 P.M. on July 4, 2012, where at that time D≈ 0.3555 and 2.8798. sin 0= cos D cos L cos + sin D sin L

Trigonometry (11th Edition)

11th Edition

ISBN:9780134217437

Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels

Chapter1: Trigonometric Functions

Section: Chapter Questions

Problem 1RE: 1. Give the measures of the complement and the supplement of an angle measuring 35°.

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Question

### Understanding the Angle of Elevation of the Sun

The angle of elevation of the sun (\(\theta\)) in the sky at any latitude (\(L\)) is calculated using the formula below. Here's how it works:

\[
\sin \theta = \cos D \cos L \cos \omega + \sin D \sin L
\]

#### Explanation of Terms:
- \(\theta = 0\) corresponds to sunrise.
- \(\theta = \frac{\pi}{2}\) occurs if the sun is directly overhead.
- \(\omega\) is the number of radians that Earth has rotated through since noon, with \(\omega = 0\) at noon.
- \(D\) is the declination of the sun, which varies because Earth is on a tilted axis.

#### Example Calculation:
Suppose we are asked to find the angle of elevation (\(\theta\)) of the sun at 11 P.M. on July 4, 2012, for Town A, where:
- The latitude \(L = 46.30^\circ\) or \(0.8081\) radian.
- At that time, \(D \approx 0.3555\) and \(\omega \approx 2.8798\).

Using the formula, we substitute the given values:
\[
\sin \theta = \cos 0.3555 \cos 0.8081 \cos 2.8798 + \sin 0.3555 \sin 0.8081
\]

Evaluating this expression gives the angle \(\theta\). This calculation involves trigonometric functions and requires a calculator or computational tool for precise results.

### Diagram Description:
There are no graphs or diagrams in the provided text.

This information aids in understanding how the angle of elevation of the sun can be calculated using specific parameters such as declination and Earth's rotation since noon.

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