**Mathematics Problem: Determining the Angle Between Routes***Context: Road Distances in Texas*Given:- The distance from Austin to Brownwood is 141 miles.- The distance from Brownwood to Corsicana is 171 miles.- The distance from Corsicana to Austin is 157 miles.Objective:Determine the angle between the routes from Brownwood using the given distances. Show your work.---**Detailed Explanation:**To find the angle between the routes from Brownwood, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. The formula for the Law of Cosines is:c² = a² + b² - 2ab * cos(C)Where:- a = 141 miles (distance from Austin to Brownwood)- b = 171 miles (distance from Brownwood to Corsicana)- c = 157 miles (distance from Corsicana to Austin)- C is the angle at Brownwood that we need to determine.Using the formula:157² = 141² + 171² - 2 * 141 * 171 * cos(C)Calculate each part:- 157² = 24649- 141² = 19881- 171² = 29241So, 24649 = 19881 + 29241 - 2 * 141 * 171 * cos(C)Combine and simplify:24649 = 49122 - 2 * 141 * 171 * cos(C)24649 - 49122 = - 2 * 141 * 171 * cos(C)-24473 = - 2 * 141 * 171 * cos(C)Divide by (-2 * 141 * 171):24473 / (2 * 141 * 171) = cos(C)24473 / 48162 ≈ cos(C)0.508 ≈ cos(C)Now, take the arccos (inverse cosine) of both sides to find C:C = arccos(0.508)C ≈ 59.4 degreesTherefore, the angle between the routes from Brownwood is approximately 59.4 degrees.

In Texas, the distance from Austin to Brownwood is 141 miles. It is 171 miles from Brownwood to Corsicana and 157 miles from Corsicana to Austin. Determine the angle between the routes from Brownwood. Show your work.

In Texas, the distance from Austin to Brownwood is 141 miles. It is 171 miles from Brownwood to Corsicana and 157 miles from Corsicana to Austin. Determine the angle between the routes from Brownwood. Show your work.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter6: The Trigonometric Functions

Section: Chapter Questions

Problem 75RE

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Concept explainers

Ratios

A ratio is a comparison between two numbers of the same kind. It represents how many times one number contains another. It also represents how small or large one number is compared to the other.

Trigonometric Ratios

Trigonometric ratios give values of trigonometric functions. It always deals with triangles that have one angle measuring 90 degrees. These triangles are right-angled. We take the ratio of sides of these triangles.

Question

**Mathematics Problem: Determining the Angle Between Routes**

*Context: Road Distances in Texas*

Given:
- The distance from Austin to Brownwood is 141 miles.
- The distance from Brownwood to Corsicana is 171 miles.
- The distance from Corsicana to Austin is 157 miles.

Objective:
Determine the angle between the routes from Brownwood using the given distances. Show your work.

---

**Detailed Explanation:**

To find the angle between the routes from Brownwood, we can use the Law of Cosines. The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles.

The formula for the Law of Cosines is:

c² = a² + b² - 2ab * cos(C)

Where:
- a = 141 miles (distance from Austin to Brownwood)
- b = 171 miles (distance from Brownwood to Corsicana)
- c = 157 miles (distance from Corsicana to Austin)
- C is the angle at Brownwood that we need to determine.

Using the formula:

157² = 141² + 171² - 2 * 141 * 171 * cos(C)

Calculate each part:
- 157² = 24649
- 141² = 19881
- 171² = 29241

So,
24649 = 19881 + 29241 - 2 * 141 * 171 * cos(C)

Combine and simplify:
24649 = 49122 - 2 * 141 * 171 * cos(C)

24649 - 49122 = - 2 * 141 * 171 * cos(C)

-24473 = - 2 * 141 * 171 * cos(C)

Divide by (-2 * 141 * 171):
24473 / (2 * 141 * 171) = cos(C)

24473 / 48162 ≈ cos(C)

0.508 ≈ cos(C)

Now, take the arccos (inverse cosine) of both sides to find C:
C = arccos(0.508)
C ≈ 59.4 degrees

Therefore, the angle between the routes from Brownwood is approximately 59.4 degrees.

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