Textbook Question
Fill in the blank(s) to correctly complete each sentence.
A triangle that is not a right triangle is a(n) _________ triangle.
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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 1
Which one of the following sets of data does not determine a unique triangle?
a. A = 50°, b = 21, a = 19
b. A = 45°, b = 10, a = 12
c. A = 130°, b = 4, a = 7
d. A = 30°, b = 8, a = 4
Verified step by step guidance1
Identify the given information for each case: angle A, side b, and side a. We want to check if these data sets determine a unique triangle.
Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). We can use this to find angle B for each set by rearranging to \(\sin B = \frac{b \sin A}{a}\).
Calculate \(\sin B\) for each set using the given values (without final numeric evaluation): \(\sin B = \frac{b \sin A}{a}\). Then analyze the possible values of angle B.
Check the range of \(\sin B\). If \(\sin B > 1\), no triangle exists. If \(0 < \sin B < 1\), there can be one or two possible angles for B (since \(\sin \theta = \sin (180^\circ - \theta)\)), which affects uniqueness.
Determine for each set whether the data leads to zero, one, or two possible triangles. The set that leads to two possible triangles does not determine a unique triangle.
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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 sides to the sines of their opposite angles in a triangle: (a/sin A) = (b/sin B) = (c/sin C). It is essential for solving triangles when given two sides and an angle not included between them (SSA), helping determine possible triangle configurations.
Recommended video:
4:27
Intro to Law of Sines
Ambiguous Case of SSA Triangles
The SSA condition can produce zero, one, or two possible triangles depending on the given measurements. This ambiguity arises because the given side opposite the known angle may or may not form a valid triangle, leading to no solution, a unique triangle, or two distinct triangles.
Recommended video:
9:50Solving SSA Triangles ("Ambiguous" Case)
Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. This property helps verify the validity of computed angles and ensures that the solutions derived from the Law of Sines or other methods correspond to a real triangle.
Recommended video:
4:47Sum and Difference of Tangent
Related Practice
Textbook Question
Use the law of sines to find the indicated part of each triangle ABC.
Find b if C = 74.2°, c = 96.3 m, B = 39.5
Textbook Question
Use the law of sines to find the indicated part of each triangle ABC.
Find B if C = 51.3°, c = 68.3 m, b = 58.2 m
Textbook Question
In each figure, a line segment of length L is to be drawn from the given point to the positive x-axis in order to form a triangle. For what value(s) of L can we draw the following?
a. two triangles
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Textbook Question
CONCEPT PREVIEW Assume a triangle ABC has standard labeling.
a. Determine whether SAA, ASA, SSA, SAS, or SSS is given.
b. Determine whether the law of sines or the law of cosines should be used to begin solving the triangle.
a, b, and C
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