Determine the number of triangles ABC possible with the given parts.


c = 50, b = 61, C = 58°

Verified step by step guidance

1. Step 1: Use the Law of Sines to find angle B. The Law of Sines states \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Substitute the known values: \( \frac{61}{\sin B} = \frac{50}{\sin 58^\circ} \).
2. Step 2: Solve for \( \sin B \) by rearranging the equation: \( \sin B = \frac{61 \cdot \sin 58^\circ}{50} \).
3. Step 3: Determine if \( \sin B \) is a valid sine value (i.e., between -1 and 1). If \( \sin B \) is greater than 1, no triangle is possible. If \( \sin B \) is less than or equal to 1, proceed to find angle B.
4. Step 4: Calculate angle B using the inverse sine function: \( B = \sin^{-1}(\sin B) \).
5. Step 5: Use the fact that the sum of angles in a triangle is 180° to find angle A: \( A = 180^\circ - C - B \). Check if the calculated angles form a valid triangle (all angles must be positive).

Key Concepts

Law of Sines

The Law of Sines relates the lengths of the sides of a triangle to the sines of its angles. It states that the ratio of a side length to the sine of its opposite angle is constant for all three sides of the triangle. This law is particularly useful for solving triangles when given two angles and a side (AAS or ASA) or two sides and a non-included angle (SSA).

Triangle Inequality Theorem

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This theorem is essential for determining the possibility of forming a triangle with given side lengths. It helps in assessing whether the given parts can indeed form a valid triangle.

Ambiguous Case of SSA

The Ambiguous Case of SSA occurs when two sides and a non-included angle are known, which can lead to zero, one, or two possible triangles. This situation arises because the given angle may not uniquely determine the triangle's shape. Understanding this case is crucial for determining how many triangles can be formed with the provided measurements.