In Exercises 1–8, solve each triangle. Round lengths of sides to the nearest tenth and angle measures to the nearest degree.

Verified step by step guidance

1. Identify the type of triangle problem you are dealing with: whether you have two sides and an included angle (SAS), two angles and a side (ASA or AAS), or three sides (SSS). This will determine which trigonometric laws or formulas to use.
2. If you have two sides and the included angle (SAS), use the Law of Cosines to find the third side: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] where \(a\) and \(b\) are known sides and \(C\) is the included angle.
3. Once you have all three sides, or if you started with three sides (SSS), use the Law of Cosines or Law of Sines to find the unknown angles. For example, to find angle \(A\) using Law of Cosines: \[ \cos(A) = \frac{b^2 + c^2 - a^2}{2bc} \]
4. If you have two angles and one side (ASA or AAS), first find the third angle by subtracting the sum of the known angles from 180°: \[ C = 180^\circ - A - B \] Then use the Law of Sines to find the unknown sides: \[ \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} \]
5. After calculating all sides and angles, round the side lengths to the nearest tenth and the angle measures to the nearest degree as required.

Key Concepts

Types of Triangles and Their Properties

Understanding whether a triangle is right, acute, or obtuse is essential because it determines which trigonometric rules apply. Recognizing side lengths and angle measures helps in choosing the correct method to solve the triangle.

Law of Sines and Law of Cosines

These laws are fundamental for solving triangles when not all sides and angles are known. The Law of Sines relates ratios of sides to the sines of opposite angles, while the Law of Cosines generalizes the Pythagorean theorem for any triangle.

Rounding and Angle Measurement Conventions

Accurate rounding of side lengths to the nearest tenth and angles to the nearest degree ensures clarity and precision in answers. Understanding degree measurement and rounding rules is crucial for presenting final solutions correctly.