In Exercises 17–32, two sides and an angle (SSA) of a triangle are given. Determine whether the given measurements produce one triangle, two triangles, or no triangle at all. Solve each triangle that results. Round to the nearest tenth and the nearest degree for sides and angles, respectively. a = 6.1, b = 4, A = 162°

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Step 1: Understand the given information. You have two sides, \(a = 6.1\) and \(b = 4\), and an angle \(A = 162^\circ\). This is an SSA (Side-Side-Angle) case, which can lead to the ambiguous case in trigonometry.

Step 2: Check if a triangle can be formed. Since \(A\) is an obtuse angle (greater than 90°), for a triangle to exist, side \(a\) must be the longest side. Verify if \(a > b\).

Step 3: Use the Law of Sines to find angle \(B\). The Law of Sines states \(\frac{a}{\sin A} = \frac{b}{\sin B}\). Rearrange to find \(\sin B = \frac{b \cdot \sin A}{a}\).

Step 4: Determine the number of possible triangles. Calculate \(\sin B\) and check if it is a valid sine value (between -1 and 1). If \(\sin B\) is valid, calculate \(B\) using \(B = \sin^{-1}(\sin B)\).

Step 5: Solve for the remaining side and angle. If a triangle is possible, use \(C = 180^\circ - A - B\) to find angle \(C\), and use the Law of Sines again to find side \(c\) with \(\frac{c}{\sin C} = \frac{a}{\sin A}\).

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주요 개념

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Law of Sines

The Law of Sines relates the ratios of the lengths of sides of a triangle to the sines of its opposite angles. It is expressed as a/b = sin(A)/sin(B) = c/sin(C). This law is particularly useful in SSA (Side-Side-Angle) cases, allowing us to determine unknown angles and sides when two sides and a non-included angle are known.

Ambiguous Case of SSA

The SSA condition can lead to an ambiguous situation where two different triangles may be formed, one triangle may be formed, or no triangle may exist at all. This ambiguity arises because the given angle may not uniquely determine the opposite side, leading to multiple possible configurations. Understanding this concept is crucial for correctly interpreting the results of the triangle's dimensions.

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the interior angles of a triangle is always 180 degrees. This theorem is essential for solving triangles, as it allows us to find unknown angles once we have determined some angles. In the context of SSA, it helps in confirming the validity of the triangles formed by ensuring that the angles add up correctly.