Solve each triangle. Approximate values to the nearest tenth.


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Verified step by step guidance

1. insert step 1: Identify the given information in the triangle, such as angles and sides.
2. insert step 2: Use the Law of Sines or Law of Cosines to find the unknown sides or angles.
3. insert step 3: If using the Law of Sines, set up the proportion $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ and solve for the unknown.
4. insert step 4: If using the Law of Cosines, use the formula $c^2=a^2+b^2-2ab\cos C$ to find the unknown side or angle.
5. insert step 5: Once all sides and angles are found, verify the solution by checking that the sum of angles is 180 degrees and the sides satisfy the triangle inequality.

Key Concepts

Triangle Types

Understanding the types of triangles—such as scalene, isosceles, and equilateral—is essential for solving triangle problems. Each type has unique properties that affect the relationships between its sides and angles, which can influence the methods used for solving them.

Trigonometric Ratios

Trigonometric ratios (sine, cosine, and tangent) relate the angles of a triangle to the lengths of its sides. These ratios are fundamental for solving triangles, especially when using the right triangle definitions or the laws of sines and cosines for non-right triangles.

Laws of Sines and Cosines

The Law of Sines and the Law of Cosines are critical for solving triangles that are not right-angled. The Law of Sines relates the ratios of sides to the sines of their opposite angles, while the Law of Cosines provides a way to find a side or angle when two sides and the included angle are known.