/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 37 A tunnel is to be dug from \(A\)... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 37

A tunnel is to be dug from \(A\) to \(B\). Both \(A\) and \(B\) are visible from \(C\). If \(A C\) is 1.4923 miles, \(B C\) is 1.0837 miles, and \(C\) is exactly \(90^{\circ}\) find the measures of angles \(A\) and \(B\)

Short Answer

Expert verified
Angle A is approximately 35.880掳 and Angle B is approximately 54.120掳.

Step by step solution

01

Understand the Triangle Configuration

Visualize the problem as a right triangle \( \triangle ABC \) where \( C \) is the right angle. The sides are \( AC = 1.4923 \) miles and \( BC = 1.0837 \) miles. The task is to find angles \( \angle A \) and \( \angle B \).
02

Use Trigonometry for Angle A

To find \( \angle A \), use the tangent function: \( \tan A = \frac{opposite}{adjacent} \). Here, \( opposite = BC = 1.0837 \) miles and \( adjacent = AC = 1.4923 \) miles. Thus, \( \tan A = \frac{1.0837}{1.4923} \).
03

Calculate Angle A

Solve for \( \angle A \) by calculating \( A = \tan^{-1} \left( \frac{1.0837}{1.4923} \right) \). Use a calculator to find the angle in degrees: \( \angle A \approx 35.880 \) degrees.
04

Find Angle B Using Angle Sum Property

Since the sum of angles in a triangle is \( 180^{\circ} \), and angle \( C = 90^{\circ} \), angle \( B \) can be found using \( B = 90^{\circ} - A \).
05

Calculate Angle B

Compute \( \angle B = 90^{\circ} - 35.880^{\circ} \approx 54.120^{\circ} \). This gives us the measure of \( \angle B \).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Trigonometric Functions
In the context of a right angle triangle, trigonometric functions play a crucial role in determining unknown angles or sides. In our problem, we're using the tangent function. The tangent of an angle in a right triangle is defined as the ratio of the length of the opposite side to the length of the adjacent side. This can be written as:
\[ \tan A = \frac{\text{opposite}}{\text{adjacent}} \]
These trigonometric functions also include sine and cosine, which relate to different pairs of a triangle's sides.
  • Sine: \( \sin A = \frac{\text{opposite}}{\text{hypotenuse}} \)
  • Cosine: \( \cos A = \frac{\text{adjacent}}{\text{hypotenuse}} \)
Understanding these functions is essential as they form the basis for angle calculations in right triangles.
Angle Calculation
Calculating angles in a right angle triangle require understanding the relationships within the triangle. In our example, we were able to find angle \( \angle A \) using the inverse tangent function. This step converts the ratio of the opposite to adjacent side into the angle itself:
\[ \angle A = \tan^{-1} \left( \frac{1.0837}{1.4923} \right) \]
This calculation uses a calculator to retrieve the angle in degrees. Once we have \( \angle A \), calculating \( \angle B \) is straightforward using the angle sum property of triangles:
  • Sum of angles in a triangle is \( 180^{\circ} \)
  • If one angle is \( 90^{\circ} \), the other two must sum to \( 90^{\circ} \)
  • \( \angle B = 90^{\circ} - \angle A \)
Calculating angles this way ensures that we have accounted for the geometry of the triangle accurately.
Right Angle Triangle
The geometry of a right triangle is distinctive because one angle is always \( 90^{\circ} \). In right triangles, the side opposite the right angle is the longest side and is called the hypotenuse. This makes right triangles especially handy in calculations involving trigonometric identities.
A right triangle like \( \triangle ABC \) here, helps us apply trigonometric ratios to find missing angles or sides by leveraging the known parts of the triangle. For example:
  • The hypotenuse is always opposite the right angle.
  • The remaining sides are referred to as the adjacent and opposite sides relative to the angle of interest.
  • Angles in the triangle will adhere to the angle sum property, forming part of the path to solving unknown measures.
This clear setup is what permits use of trigonometric ratios efficiently.
Geometry
Geometry is the branch of mathematics dealing with the properties and relations of points, lines, surfaces, and solids. In our exercise, we explore the geometric nature of triangles, focusing on a right triangle.
Geometry offers us principles like the angle sum property, which is crucial for validating calculations. The triangle's configuration and measurement precision allow us to utilize trigonometric functions to solve complex problems.
  • Geometric principles ensure that our mathematical models are both accurate and reliable.
  • Visualization is often a helpful technique in geometry to better understand and solve problems.
  • Using the properties of shapes like triangles can simplify even seemingly complicated tasks.
This harmony between geometry and trigonometry forms a powerful toolset for understanding and solving problems efficiently.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Use the appropriate reciprocal identity to find each function value. Rationalize denominators when applicable. $$\tan \theta, \text { given that } \cot \theta=-0.01$$

The first accurate estimate of the distance around Earth was done by the Greek astronomer Eratosthenes \((276-195 \text { B.C.) },\) who noted that the noontime position of the sun at the summer solstice differed by \(7^{\circ} 12^{\prime}\) from the city of Syene to the city of Alexandria. (See the figure.) The distance between these two cities is 496 miles. Use the are length formula to estimate the radius of Earth. Then approximate the circumference of Earth.

The speedometer of a pickup truck is designed to be accurate with tires of radius 14 inches. (a) Find the number of rotations, to the nearest whole number, of a tire in 1 hour if the truck is driven \(55 \mathrm{mph}\) (b) Suppose that oversize tires of radius 16 inches are placed on the truck. If the truck is now driven for 1 hour with the speedometer reading 55 mph, how far has the truck gone? If the speed limit is \(60 \mathrm{mph}\) did the driver exceed the speed limit?

Use fundamental identities to find each expression. Write \(\sin \theta\) in terms of \(\sec \theta\) if \(\theta\) is in quadrant I or II.

If \(n\) is an integer, \(n \cdot 180^{\circ}\) represents an integer multiple of \(180^{\circ},(2 n+1) \cdot 90^{\circ}\) represents an odd integer multiple of \(90^{\circ}\), and so on. Decide whether each expression is equal to \(0,1\), or \(-1\) or is undefined. $$\sin \left(n \cdot 180^{\circ}\right)$$
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Mechanics Maths

Read Explanation

Calculus

Read Explanation

Discrete Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.