What is the cosine of an angle in a right-angled triangle?

The cosine of an angle in a right-angled triangle is the ratio of the adjacent side to the hypotenuse.

In a right-angled triangle, the cosine (often abbreviated as "cos") of one of the non-right angles is a way to relate the lengths of the sides of the triangle. Specifically, for an angle θ, the cosine is calculated by dividing the length of the side adjacent to θ by the length of the hypotenuse. Mathematically, this is expressed as:

\[ \cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}} \]

The hypotenuse is always the longest side of the right-angled triangle, opposite the right angle. The adjacent side is the one that forms the angle θ with the hypotenuse, but it is not the hypotenuse itself.

For example, if you have a right-angled triangle where the length of the adjacent side to angle θ is 3 units and the hypotenuse is 5 units, the cosine of angle θ would be:

\[ \cos(\theta) = \frac{3}{5} = 0.6 \]

Understanding the cosine function is crucial for solving various problems in trigonometry, such as finding missing side lengths or angles in right-angled triangles. It also has applications in different fields like physics, engineering, and even computer graphics. Remember, the cosine function is just one of the three primary trigonometric ratios, the others being sine (sin) and tangent (tan), which relate the sides of a right-angled triangle in different ways.