In oblique triangle ABC, C = 68°, a = 5, and b = 6. Find c to the nearest tenth.

Verified step by step guidance

1. Identify the given elements in triangle ABC: angle C = 68°, side a = 5 (opposite angle A), and side b = 6 (opposite angle B). We need to find side c (opposite angle C).
2. Use the Law of Cosines formula to find side c, which relates the sides and the included angle: \(c^2 = a^2 + b^2 - 2ab \cdot \cos(C)\).
3. Substitute the known values into the Law of Cosines formula: \(c^2 = 5^2 + 6^2 - 2 \times 5 \times 6 \times \cos(68^\circ)\).
4. Calculate the right-hand side expression step-by-step: square the sides, multiply the terms, and find the cosine of 68° (using a calculator or trigonometric table).
5. Take the square root of the result to find side c: \(c = \sqrt{\text{calculated value}}\). Round the answer to the nearest tenth as requested.

Key Concepts

Law of Cosines

The Law of Cosines relates the lengths of sides of any triangle to the cosine of one of its angles. It is especially useful in oblique triangles where no right angle is present. The formula is c² = a² + b² - 2ab cos(C), allowing calculation of the unknown side when two sides and the included angle are known.

Oblique Triangle

An oblique triangle is any triangle that does not contain a right angle. Solving oblique triangles often requires the Law of Sines or Law of Cosines, as the Pythagorean theorem does not apply. Understanding the classification helps determine the appropriate method for finding unknown sides or angles.

Angle-Side Relationship

In any triangle, the side lengths and angles are interdependent; knowing two sides and the included angle allows calculation of the third side. This relationship is fundamental in applying the Law of Cosines, ensuring the correct angle is used between the given sides to find the unknown side length.