80. A group of campers hikes down a steep path. One member of the group has an altimeter on his watch to measure altitude. If the path is 1250 yd and the amount of altitude lost is 480 yd, what is the angle of incline? Round to the nearest tenth of a degree. 1250 yd 480 yd
80. A group of campers hikes down a steep path. One member of the group has an altimeter on his watch to measure altitude. If the path is 1250 yd and the amount of altitude lost is 480 yd, what is the angle of incline? Round to the nearest tenth of a degree. 1250 yd 480 yd
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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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
![### Educational Content
#### Right Triangle Trigonometry Problems
**Problem 80:**
A group of campers hikes down a steep path. One member of the group has an altimeter on his watch to measure altitude. If the path is 1250 yards and the amount of altitude lost is 480 yards, what is the angle of incline? Round to the nearest tenth of a degree.
*Diagram Explanation:*
- There is a right triangle shown.
- The hypotenuse (the path the hikers travel) is labeled 1250 yards.
- The opposite side (amount of altitude lost) is labeled 480 yards.
- The angle of inclination, labeled \(\theta\), is at the bottom of the triangle opposite the 480-yard side.
**Solution Steps:**
1. Use trigonometric relationships (specifically sine, since it involves opposite and hypotenuse):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{480}{1250} \]
2. Calculate the angle \(\theta\):
\[ \theta = \sin^{-1} \left( \frac{480}{1250} \right) \]
3. Plug in the values and round to the nearest tenth of a degree.
**Problem 82:**
A ski run on Giant Steps Mountain in Utah is 1475 meters long. The difference in altitude from the beginning to the end of the run is 350 meters. Find the angle of the ski run. Round to the nearest tenth of a degree.
*Solution Steps:*
1. Use trigonometric relationships (specifically sine):
\[ \sin(\theta) = \frac{350}{1475} \]
2. Calculate the angle \(\theta\):
\[ \theta = \sin^{-1} \left( \frac{350}{1475} \right) \]
3. Plug in the values and round to the nearest tenth of a degree.
**For Exercises 111–114:**
Use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write \(\theta\) as a function of \(x\) in three different ways. It is not necessary to rationalize the denominator.
**Problem 112:**
*Diagram Explanation:*
- There is a right triangle.
- One leg (adjacent to \(\theta\)) is labeled 2.
- The hyp](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F231fae22-ac62-4382-9eb0-95d912af77d1%2F09d7cd75-682a-41b7-928e-ed0112bd0438%2Fe5nma7_reoriented.jpeg&w=3840&q=75)
Transcribed Image Text:### Educational Content
#### Right Triangle Trigonometry Problems
**Problem 80:**
A group of campers hikes down a steep path. One member of the group has an altimeter on his watch to measure altitude. If the path is 1250 yards and the amount of altitude lost is 480 yards, what is the angle of incline? Round to the nearest tenth of a degree.
*Diagram Explanation:*
- There is a right triangle shown.
- The hypotenuse (the path the hikers travel) is labeled 1250 yards.
- The opposite side (amount of altitude lost) is labeled 480 yards.
- The angle of inclination, labeled \(\theta\), is at the bottom of the triangle opposite the 480-yard side.
**Solution Steps:**
1. Use trigonometric relationships (specifically sine, since it involves opposite and hypotenuse):
\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{480}{1250} \]
2. Calculate the angle \(\theta\):
\[ \theta = \sin^{-1} \left( \frac{480}{1250} \right) \]
3. Plug in the values and round to the nearest tenth of a degree.
**Problem 82:**
A ski run on Giant Steps Mountain in Utah is 1475 meters long. The difference in altitude from the beginning to the end of the run is 350 meters. Find the angle of the ski run. Round to the nearest tenth of a degree.
*Solution Steps:*
1. Use trigonometric relationships (specifically sine):
\[ \sin(\theta) = \frac{350}{1475} \]
2. Calculate the angle \(\theta\):
\[ \theta = \sin^{-1} \left( \frac{350}{1475} \right) \]
3. Plug in the values and round to the nearest tenth of a degree.
**For Exercises 111–114:**
Use the relationship given in the right triangle and the inverse sine, cosine, and tangent functions to write \(\theta\) as a function of \(x\) in three different ways. It is not necessary to rationalize the denominator.
**Problem 112:**
*Diagram Explanation:*
- There is a right triangle.
- One leg (adjacent to \(\theta\)) is labeled 2.
- The hyp
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