Trigonometric Functions of Real Numbers

The unit circle—a circle with a radius of one, centered at the origin of the coordinate plane—serves as the foundational framework for defining trigonometric functions. In this context, arc length refers to the distance measured along the circumference of the circle between two points, and it provides a way to represent real numbers geometrically. Each real number t corresponds to an arc length measured counterclockwise from the positive x-axis around the circle. The coordinates of a point on the unit circle (x, y) at a given arc length define the cosine and sine of the real number, respectively, with x = cos⁡ t and y = sin⁡ t.

Other trigonometric functions are derived directly from sine and cosine. The tangent function is defined as the ratio of sine to cosine:

Conversely, the cotangent function is the ratio of cosine to sine:

The secant and cosecant functions are the reciprocals of cosine and sine, respectively:

Dividing the unit circle into four quadrants clarifies the signs of the trigonometric functions. In the first quadrant, all functions are positive. In the second quadrant, only sine and cosecant retain positive values. The third quadrant renders tangent and cotangent positive, while cosine and secant are positive in the fourth quadrant. These sign rules are essential for evaluating trigonometric functions across different angles.

The symmetry of the unit circle imparts even or odd properties to the functions. Cosine and secant are even functions, satisfying cos⁡(−t) = cos⁡ t and sec⁡(−t) = sec⁡ t. In contrast, sine, tangent, cotangent, and cosecant are odd functions, satisfying identities such as sin⁡(−t) = −sin⁡ t and tan⁡(−t) = −tan⁡ t.

Together, these definitions and sign conventions offer a coherent framework for understanding the behavior of trigonometric functions. The unit circle not only provides geometric intuition but also facilitates precise calculation and analysis, making it indispensable in both pure and applied mathematics.