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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 7.10
Chapter 8, Problem 7.10

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 guidance
1
Identify the given information: three sides of the triangle are known, which means we have a Side-Side-Side (SSS) scenario.
Recall that the Law of Sines is typically used when we have an Angle-Side-Angle (ASA), Angle-Angle-Side (AAS), or Side-Angle-Side (SAS) scenario, not SSS.
Understand that for a triangle with three known sides, the Law of Cosines is more appropriate to find the angles.
Use the Law of Cosines to find one of the angles. For example, to find angle A, use the formula: \( \cos A = \frac{b^2 + c^2 - a^2}{2bc} \).
Once one angle is found using the Law of Cosines, the Law of Sines can be used to find the remaining angles if needed.

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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 states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant for all three sides and angles. This relationship is expressed as a/b = sin(A)/sin(B) = c/sin(C). It is particularly useful for solving triangles when we have either two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA).
Recommended video:
4:27
Intro to Law of Sines

Triangle Properties

A triangle is defined by three sides and three angles, and the sum of the angles in any triangle is always 180 degrees. When three sides are known, the triangle can be solved using the Law of Cosines to find the angles first, which can then be used with the Law of Sines if needed. Understanding these properties is crucial for determining the feasibility of solving a triangle.
Recommended video:
4:42
Review of Triangles

Ambiguous Case of SSA

The SSA (Side-Side-Angle) condition can lead to an ambiguous situation where two different triangles may be formed, one triangle, or no triangle at all. This ambiguity arises because knowing two sides and a non-included angle does not guarantee a unique triangle. Recognizing this case is essential when applying the Law of Sines to ensure accurate solutions.
Recommended video:
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Solving SSA Triangles ("Ambiguous" Case)
Related Practice
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

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.


(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

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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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Textbook Question

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

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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

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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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