Textbook Question
Solve each triangle ABC that exists.
B = 72.2°, b = 78.3 m, c = 145 m
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7. Non-Right Triangles
Law of Sines
Problem 3
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
Verified step by step guidance1
Identify the known parts of the triangle: angle \(C = 51.3^\circ\), side \(c = 68.3\) m (opposite angle \(C\)), and side \(b = 58.2\) m (opposite angle \(B\)). We need to find angle \(B\).
Recall the Law of Sines formula: \(\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}\). We will use the relationship between sides \(b\), \(c\) and angles \(B\), \(C\).
Set up the equation using the Law of Sines for sides \(b\) and \(c\): \(\frac{b}{\sin B} = \frac{c}{\sin C}\). Substitute the known values: \(\frac{58.2}{\sin B} = \frac{68.3}{\sin 51.3^\circ}\).
Solve for \(\sin B\) by cross-multiplying: \(\sin B = \frac{58.2 \times \sin 51.3^\circ}{68.3}\). Calculate the right side to find \(\sin B\) (do not compute the final value here).
Find angle \(B\) by taking the inverse sine (arcsin) of the value obtained for \(\sin B\): \(B = \sin^{-1}(\sin B)\). Remember to consider the possible ambiguous case for angle \(B\) in 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 states that (a/sin A) = (b/sin B) = (c/sin C). This law is useful for solving triangles when given two angles and one side or two sides and a non-included angle.
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Triangle Angle Sum Property
The sum of the interior angles in any triangle is always 180°. Knowing two angles allows you to find the third by subtracting their sum from 180°. This property helps verify or find missing angles after applying the Law of Sines.
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Solving for an Unknown Angle
When two sides and an angle opposite one of them are known, the Law of Sines can be used to find an unknown angle by rearranging the formula and using inverse sine functions. Care must be taken to consider the ambiguous case where two different angles may satisfy the equation.
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