Angles and Their Measurement

In trigonometry, an angle is generated by the rotation of a ray from an initial position (the initial side) to a final position (the terminal side), both sharing a common starting point called the vertex. Business and statistical applications of trigonometry — for example, modelling seasonal or cyclical patterns in sales figures or index numbers — need a precise, unambiguous way of measuring such angles, and two systems are in common use.

Sexagesimal (degree) measure. Here a complete rotation is divided into $360$ equal parts, each called one degree ($1^\circ$). Each degree is further divided into $60$ minutes ($1^\circ = 60'$), and each minute into $60$ seconds ($1' = 60''$). This is the system most familiar from everyday use.

Radian measure. A radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the radius of that circle. Because the circumference of a circle of radius $r$ is $2\pi r$, a complete rotation subtends an angle of $2\pi$ radians. This gives the fundamental relation

from which every conversion follows:

Angles in standard position. An angle is said to be in standard position when its vertex is placed at the origin of a coordinate system and its initial side lies along the positive $x$-axis. The angle is taken as positive when the terminal side is reached by rotating anticlockwise, and negative when reached by rotating clockwise. This convention is what makes it possible to talk about trigonometric ratios of angles larger than $360^\circ$ or of negative angles, which becomes essential once periodic business cycles are studied later in the course.