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
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7. Non-Right Triangles
Law of Sines
Problem 5
Textbook Question
Determine the number of triangles ABC possible with the given parts.
a = 50, b = 26, A = 95°
Verified step by step guidance1
Identify the given elements: side \(a = 50\), side \(b = 26\), and angle \(A = 95^\circ\). Note that angle \(A\) is opposite side \(a\).
Use the Law of Sines to find angle \(B\). The Law of Sines states: \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Substitute the known values to get \(\frac{50}{\sin 95^\circ} = \frac{26}{\sin B}\).
Rearrange the equation to solve for \(\sin B\): \(\sin B = \frac{26 \times \sin 95^\circ}{50}\).
Evaluate \(\sin B\) and determine the possible values of angle \(B\). Since \(\sin B\) can correspond to two different angles between \(0^\circ\) and \(180^\circ\) (one acute and one obtuse), consider both possibilities for \(B\).
For each possible value of \(B\), calculate angle \(C\) using the triangle angle sum property: \(C = 180^\circ - A - B\). Check if \(C\) is positive and valid. The number of valid triangles corresponds to the number of valid \(C\) values found.
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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 sides and angles of a triangle through the ratio a/sin A = b/sin B = c/sin C. It is essential for solving triangles when given two sides and an angle, especially to find unknown angles or sides.
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Ambiguous Case of SSA Triangles
When two sides and a non-included angle (SSA) are given, there can be zero, one, or two possible triangles. This ambiguity arises because the given angle and side lengths may correspond to different triangle configurations.
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Triangle Inequality and Angle-Side Relationships
The triangle inequality states that the sum of any two sides must be greater than the third. Also, larger angles face longer sides. These principles help determine if a triangle is possible and how many distinct triangles can be formed.
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