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How are trigonometric functions represented on the unit circle?

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Diagram of a right triangle labeled with sides a, b, and hypotenuse c, alongside formulas for the six trigonometric functions-sine, cosine, tangent, cosecant, secant, and cotangent-each defined by ratios of the triangle's sides. — trigonometry Based on the definitions, various simple relationships exist among the functions. For example, csc A = 1/sin A, sec A = 1/cos A, cot A = 1/tan A, and tan A = sin A/cos A.

In trigonometry, the unit circle—a circle with a radius of 1 centered at the origin (0,0)—provides a geometric framework for defining trigonometric functions beyond right triangles.12

When a line is drawn from the origin at an angle $A$ with respect to the $x$ -axis, it intersects the unit circle at a point $(x, y)$ . The trigonometric functions are represented as follows:

Cosine and Sine: The $x$ -coordinate of the intersection point corresponds to the cosine of the angle, while the $y$ -coordinate corresponds to the sine function.234
Tangent: The tangent is represented as the slope of the line, which can be calculated as the ratio of the coordinates: $\tan A = \frac{y}{x}$ .5
Cotangent: The cotangent is the reciprocal of the slope of this line.1

These circular functions allow for the extension of trigonometric concepts to all real numbers, forming a foundation for mathematical analysis.2