In Exercises 9–16, solve each triangle. Round lengths to the nearest tenth and angle measures to the nearest degree.A = 85°, B = 35°, c = 30

Verified step by step guidance

1. insert step 1: Use the fact that the sum of angles in a triangle is 180° to find angle C. Calculate C = 180° - A - B.
2. insert step 2: Apply the Law of Sines to find side a. The Law of Sines states that \( \frac{a}{\sin A} = \frac{c}{\sin C} \). Rearrange to solve for a: \( a = c \cdot \frac{\sin A}{\sin C} \).
3. insert step 3: Use the Law of Sines again to find side b. The Law of Sines states that \( \frac{b}{\sin B} = \frac{c}{\sin C} \). Rearrange to solve for b: \( b = c \cdot \frac{\sin B}{\sin C} \).
4. insert step 4: Calculate the sine of each angle using a calculator or trigonometric tables.
5. insert step 5: Substitute the known values into the equations for a and b to find their lengths, rounding to the nearest tenth.

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, the ratio of a side length to the sine of its opposite angle is constant. This law is particularly useful for solving triangles when two angles and one side (AAS or ASA) or two sides and a non-included angle (SSA) are known.

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 finding unknown angles in a triangle when two angles are known. In the given problem, knowing angles A and B allows us to calculate angle C by subtracting the sum of A and B from 180 degrees.

Rounding Rules

Rounding rules are guidelines for approximating numbers to a specified degree of accuracy. In this context, lengths are rounded to the nearest tenth and angle measures to the nearest degree. Understanding how to round correctly is crucial for providing answers that meet the problem's requirements, ensuring that the final results are both accurate and presented in the correct format.