Circular Definitions

Trigonometric functions (also called circular functions) are defined using the unit circle, $x^2+y^2=1.$ Let $\theta$ be a central angle, measured counterclockwise, subtended by the arc with initial point $(1,0)$ and endpoint $(x,y).$ The sine of $\theta,$ denoted $\sin{\theta},$ is defined as the vertical component of the endpoint of the arc. The cosine of $\theta,$ denoted $\cos{\theta},$ is defined as the horizontal component. The coordinates $(x,y)$ of any point on the unit circle therefore correspond to $(\cos{\theta}, \sin{\theta}).$

The ratio $\sin{\theta}/\cos{\theta}$ is defined as the tangent of $\theta,$ denoted $\tan{\theta}.$

The reciprocal of the sine of $\theta$ is defined as the cosecant of $\theta,$ denoted $\csc{\theta}.$ The reciprocal of the cosine of $\theta$ is defined as the secant of $\theta,$ denoted $\sec{\theta}.$ The reciprocal of the tangent of $\theta$ is defined as the cotangent of $\theta,$ denoted $\cot{\theta}.$

An obvious consequence of these ``circular'' definitions is the trigonometric functions are periodic with period $2\pi.$ That is,

func$$(2\pi n+\theta)=$$func$$(\theta),$$

where $n$ is an integer and func is a trigonometric function. Additionally, The tangent and cotangent functions have period $\pi.$