Cosine

The cosine of an angle in a right-angled triangle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse. It is one of the basic trigonometric functions and is denoted by the abbreviation "cos". The cosine function is widely used in mathematics, physics, engineering, and many other fields.

Definition

The cosine of an angle θ in a right-angled triangle is calculated using the following formula:

cos(θ) = adjacent / hypotenuse

Important Points to Remember

1. The value of the cosine function ranges from -1 to 1.
2. The cosine function is an even function, which means that cos(θ) = cos(-θ).
3. The cosine function has a period of 360 degrees (or 2π radians), which means the function repeats itself every 360 degrees or 2π radians.

Graph of the Cosine Function

The graph of the cosine function is a periodic wave that oscillates between -1 and 1. The general shape of the graph is a smooth, continuous curve that repeats itself every 360 degrees (or 2π radians).

Common Cosine Values

Practice Problems

1. Find the cosine of a 45-degree angle in a right-angled triangle with an adjacent side of length 4 and a hypotenuse of length 5.
2. Graph the function y = cos(x) for x in the interval [0, 2π].
3. Determine the value of θ if cos(θ) = -1.

Conclusion

The cosine function is a fundamental concept in trigonometry and has wide-ranging applications in various fields. Understanding the properties and behavior of the cosine function is essential for solving problems involving angles and right-angled triangles.