In Exercises 1–12, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree. If no triangle exists, state 'no triangle.' If two triangles exist, solve each triangle. C = 50°, a = 3, c = 1

Verified step by step guidance

1. Step 1: Use the Law of Sines to find angle A. The Law of Sines states that \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Substitute the known values: \( \frac{3}{\sin A} = \frac{1}{\sin 50^\circ} \).
2. Step 2: Solve for \( \sin A \) by rearranging the equation: \( \sin A = \frac{3 \cdot \sin 50^\circ}{1} \). Calculate \( \sin A \) using the sine of 50 degrees.
3. Step 3: Determine if \( \sin A \) is a valid sine value (i.e., between -1 and 1). If \( \sin A > 1 \), then no triangle exists. If \( \sin A \leq 1 \), proceed to find angle A using \( A = \sin^{-1}(\sin A) \).
4. Step 4: Calculate angle B using the angle sum property of triangles: \( B = 180^\circ - A - C \).
5. Step 5: Use the Law of Sines again to find side b: \( \frac{b}{\sin B} = \frac{a}{\sin A} \). Rearrange to solve for b: \( b = \frac{a \cdot \sin B}{\sin A} \).

Key Concepts

Law of Sines

The Law of Sines is a fundamental principle in trigonometry that relates the ratios of the lengths of sides of a triangle to the sines of its angles. It states that for any triangle with sides a, b, and c opposite angles A, B, and C respectively, the ratio a/sin(A) = b/sin(B) = c/sin(C) holds true. This law is particularly useful for solving triangles when given two angles and one side or two sides and a non-included angle.

Ambiguous Case of the Law of Sines

The ambiguous case occurs when using the Law of Sines to solve for a triangle with two sides and a non-included angle (SSA condition). In this scenario, there can be zero, one, or two possible triangles depending on the given measurements. Understanding this concept is crucial for determining whether a triangle can be formed and how many solutions exist.

Triangle Existence Criteria

Triangle existence criteria dictate the conditions under which a triangle can be formed from given side lengths and angles. For example, the sum of the angles in a triangle must equal 180 degrees, and the lengths of any two sides must be greater than the length of the third side. These criteria help in identifying whether a triangle can exist based on the provided dimensions.