Textbook Question
Solve each triangle. See Examples 2 and 3.
a = 3.0 ft, b = 5.0 ft, c = 6.0 ft
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Ch. 7 - Applications of Trigonometry and Vectors
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 8, Problem 34
To determine the distance RS across a deep canyon, Rhonda lays off a distance TR = 582 yd. She then finds that T = 32° 50' and R = 102° 20'. Find RS. See the figure.
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Verified step by step guidance1
Identify the triangle involved and label the vertices as T, R, and S, where TR is the known side with length 582 yards, and the angles at T and R are given as 32° 50' and 102° 20' respectively.
Convert the given angles from degrees and minutes to decimal degrees for easier calculation: for example, 32° 50' becomes 32 + 50/60 degrees.
Calculate the third angle at vertex S using the triangle angle sum property: \(\angle S = 180^\circ - \angle T - \angle R\).
Use the Law of Sines to find the length of side RS. The Law of Sines states: \(\frac{RS}{\sin(\angle T)} = \frac{TR}{\sin(\angle S)}\).
Rearrange the Law of Sines formula to solve for RS: \(RS = \frac{TR \times \sin(\angle T)}{\sin(\angle S)}\). Substitute the known values and calculate the sine values to find RS.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Law of Sines
The Law of Sines relates the sides and angles of any triangle, stating that the ratio of a side length to the sine of its opposite angle is constant. It is essential for finding unknown sides or angles when two angles and one side are known, as in this problem involving triangle measurements.
Recommended video:
4:27
Intro to Law of Sines
Angle Measurement and Conversion
Understanding how to interpret and convert angles given in degrees and minutes is crucial. Since angles are provided in degrees and minutes (e.g., 32° 50'), converting them to decimal degrees or radians facilitates calculations using trigonometric functions.
Recommended video:
5:31Reference Angles on the Unit Circle
Triangle Angle Sum Property
The sum of the interior angles of any triangle is always 180°. Knowing two angles allows calculation of the third, which is necessary to apply the Law of Sines correctly and solve for the unknown side RS in the triangle.
Recommended video:
4:47Sum and Difference of Tangent
Related Practice
Textbook Question
A ship is sailing due north. At a certain point the bearing of a lighthouse 12.5 km away is N 38.8° E. Later on, the captain notices that the bearing of the lighthouse has become S 44.2° E. How far did the ship travel between the two observations of the lighthouse?
Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u
Textbook Question
Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.
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Textbook Question
Apply the law of sines to the following:
A = 104°, a = 26.8, b = 31.3.
What happens when we try to find the measure of angle B using a calculator?
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Textbook Question
Two forces of 692 newtons and 423 newtons act on a point. The resultant force is 786 newtons. Find the angle between the forces.
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