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

Verified step by step guidance

1. Identify that you are dealing with an oblique triangle, which means it is not a right triangle.
2. Use the Law of Cosines to find the length of side c. The Law of Cosines states: \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).
3. Substitute the given values into the formula: \( c^2 = 5^2 + 6^2 - 2 \cdot 5 \cdot 6 \cdot \cos(68^\circ) \).
4. Calculate the values of \( 5^2 \) and \( 6^2 \), and then compute \( 2 \cdot 5 \cdot 6 \cdot \cos(68^\circ) \).
5. Solve for \( c \) by taking the square root of the result from the previous step to find the length of side c to the nearest tenth.

Key Concepts

Law of Cosines

The Law of Cosines relates the lengths of the sides of a triangle to the cosine of one of its angles. It is particularly useful in oblique triangles where the angle and two sides are known. The formula is c² = a² + b² - 2ab * cos(C), allowing us to find the length of the third side when two sides and the included angle are given.

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the ratios of its sides. In this context, cosine is used to find the length of side c based on the known angle C and the lengths of sides a and b. Understanding these functions is essential for solving problems involving angles and side lengths in triangles.

Angle Measurement

Angle measurement is crucial in trigonometry, as it determines the relationships between the sides of a triangle. Angles can be measured in degrees or radians, and in this problem, angle C is given in degrees. Accurate angle measurement is necessary for applying trigonometric laws and functions effectively to find unknown side lengths.