Textbook Question
Find the length of the remaining side of each triangle. Do not use a calculator.
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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 10
Consider each case and determine whether there is sufficient information to solve the triangle using the law of sines.
Three sides are known.
Verified step by step guidance1
Recall that the Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\), where \(a\), \(b\), and \(c\) are the sides opposite angles \(A\), \(B\), and \(C\) respectively.
When three sides of a triangle are known (SSS case), the Law of Sines alone is not sufficient to solve the triangle because it relates sides to angles but does not directly provide an angle without at least one angle known.
Instead, use the Law of Cosines to find one angle first. The Law of Cosines formula is: \(c^2 = a^2 + b^2 - 2ab \cos C\) (and similarly for other angles).
After finding one angle using the Law of Cosines, you can then apply the Law of Sines to find the remaining angles and sides if needed.
Therefore, with three sides known, the Law of Sines alone is not sufficient; you must start with the Law of Cosines to solve the 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 the lengths of sides of a triangle to the sines of their opposite angles. It is expressed as (a/sin A) = (b/sin B) = (c/sin C). This law is useful for solving triangles when given certain combinations of sides and angles, but it requires at least one angle-side opposite pair.
Recommended video:
4:27
Intro to Law of Sines
Triangle Solving Criteria
To solve a triangle, sufficient information about sides and angles must be known. The Law of Sines is applicable when at least one angle and its opposite side are known. Knowing only three sides (SSS) does not provide an angle-side pair, so the Law of Sines alone cannot solve the triangle in this case.
Recommended video:
5:19Solving Right Triangles with the Pythagorean Theorem
Law of Cosines
The Law of Cosines is used to solve triangles when three sides are known (SSS) or two sides and the included angle (SAS). It relates the lengths of sides to the cosine of an angle, allowing calculation of angles from side lengths. For three known sides, the Law of Cosines is the appropriate method, not the Law of Sines.
Recommended video:
4:35Intro to Law of Cosines
Related Practice
Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈-4, -7〉
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Textbook Question
Determine the number of triangles ABC possible with the given parts.
c = 50, b = 61, C = 58°
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Textbook Question
Find the magnitude and direction angle for each vector. Round angle measures to the nearest tenth, as necessary.
〈15, -8〉
686
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Textbook Question
Refer to vectors a through h below. Make a copy or a sketch of each vector, and then draw a sketch to represent each of the following. For example, find a + e by placing a and e so that their initial points coincide. Then use the parallelogram rule to find the resultant, as shown in the figure on the right.
a + b
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Textbook Question
Find each angle B. Do not use a calculator.
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