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The Law of Cosines is a fundamental theorem in trigonometry that extends the Pythagorean Theorem to all triangles. It relates the lengths of a triangle's sides to the cosine of one of its angles, enabling the calculation of unknown side lengths and angles in non-right-angled triangles. This theorem is crucial in fields such as navigation, construction, and physics, where precise geometric measurements are necessary.

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## Definition and Importance

### Essential theorem in trigonometry

The Law of Cosines generalizes the Pythagorean Theorem to include all types of triangles

### Instrumental in computing triangle side lengths

Formula for solving oblique triangles

The Law of Cosines is particularly valuable for solving oblique triangles, which do not contain a right angle

### Practical applications

The Law of Cosines has practical applications in fields such as navigation, construction, and physics

## Derivation and Variations

### Derived from Pythagorean Theorem and Cosine Ratio

The Law of Cosines can be derived by applying the Pythagorean Theorem and incorporating the Cosine Ratio

### Three variations of the formula

Solving for side lengths

The Law of Cosines can be adapted to solve for any side length within a triangle

Solving for unknown angles

The Law of Cosines can also be rearranged to solve for unknown angles within a triangle

## Practical Applications

### Distance calculation

The Law of Cosines can be used to calculate the distance between two points when certain conditions are met

### Solving oblique triangles

The Law of Cosines is a powerful tool for solving oblique triangles in mathematics, engineering, and other technical disciplines

## Solving Triangles

### Necessary information for solving a triangle

To solve a triangle with the Law of Cosines, one must know at least one side's length and two additional elements (sides or angles) of the triangle

### Choosing the appropriate method

The Law of Cosines is the appropriate method when two sides and the included angle, or all three sides, are known