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
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?
Verified step by step guidance1
Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B}\), where \(A\) and \(B\) are angles opposite sides \(a\) and \(b\) respectively.
Substitute the known values into the formula: \(\frac{26.8}{\sin 104^\circ} = \frac{31.3}{\sin B}\).
Rearrange the equation to solve for \(\sin B\): \(\sin B = \frac{31.3 \times \sin 104^\circ}{26.8}\).
Use a calculator to evaluate the right-hand side expression to find the numerical value of \(\sin B\).
Observe the result: if the value of \(\sin B\) is greater than 1, this indicates that no angle \(B\) exists with such a sine value, meaning the triangle cannot be formed with the given measurements.
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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 ratios of the lengths of sides of a triangle to the sines of their opposite angles: (a/sin A) = (b/sin B) = (c/sin C). It is used to find unknown sides or angles in any triangle when given sufficient information, especially in non-right triangles.
Recommended video:
4:27
Intro to Law of Sines
Ambiguous Case in the Law of Sines
When using the Law of Sines to find an angle, the inverse sine function can yield two possible angle measures (an acute and an obtuse angle) because sin(θ) = sin(180° - θ). This ambiguity can cause confusion or multiple solutions, especially in SSA (Side-Side-Angle) configurations.
Recommended video:
9:50Solving SSA Triangles ("Ambiguous" Case)
Using a Calculator for Inverse Sine
Calculators typically return the principal value of the inverse sine function, which is an angle between -90° and 90°. When solving triangles, this means the calculator gives only one possible angle, so you must consider the possibility of a second valid angle (180° minus the calculator’s result) to fully solve the problem.
Recommended video:
4:03Inverse Sine
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?
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Textbook Question
Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
v - u
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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
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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Textbook Question
Without using the law of sines, explain why no triangle ABC can exist that satisfies A = 103° 20', a = 14.6 ft, b = 20.4 ft.
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