Trigonometry is the branch of mathematics that studies triangles and their relationships. It explores the connections between angles and sides in triangles, particularly right triangles. The three main trigonometric functions - sine, cosine, and tangent - are fundamental ratios that help us understand these relationships. These functions are essential in mathematics, physics, and engineering applications.
A right triangle has three fundamental components. The hypotenuse is the longest side, always opposite the right angle. The opposite side is across from the angle we're considering, while the adjacent side is next to that angle. These sides change their roles depending on which angle we focus on. Understanding these relationships is crucial for defining trigonometric functions.
The sine function is the first fundamental trigonometric ratio. It is defined as the opposite side divided by the hypotenuse. This ratio remains constant for any given angle, regardless of the triangle's size. For example, the sine of 30 degrees always equals one half. We can demonstrate this with triangles of different sizes but the same angle - the sine ratio stays the same.
The cosine function is defined as the adjacent side divided by the hypotenuse. Cosine has a complementary relationship with sine - the sine of an angle equals the cosine of its complement. In a 30-60-90 triangle, cosine 30 degrees equals square root of 3 over 2, while cosine 60 degrees equals one half. Notice how the adjacent side changes depending on which angle we consider.
The tangent function completes our trio of basic trigonometric functions. It is defined as the opposite side divided by the adjacent side, and can also be expressed as sine divided by cosine. Tangent represents the slope of the angle. In a 45-degree right triangle, where both legs are equal, tangent 45 degrees equals 1. Unlike sine and cosine, tangent values can exceed 1 and approach infinity as the angle approaches 90 degrees.