Overview
Trigonometry is defined by the measurement of angles and their relationships. One of the ways that trigonometry can be applied is in the measurement of angles and circular motion.
Angle Measure
Angles in trigonometry and calculus can be measured in radians, which is a relationship of the measurement of an angle by the arc it makes on a circle of radius 1. If the initial side of the ray is measured counterclockwise from the terminal side, it is a positive angle, and if the initial side of the ray is measured clockwise from the terminal side, it is a negative angle.
Standard Position
The standard position for measuring an angle θ is to measure it with the vertex at the origin at the xy axis and the initial side of the angle along the x axis. This is the same concept as the angle at the unit circle. If two angles are both in the standard position, and both sides coincide, they are coterminal. The initial side of one angle is the terminal side of the other and the terminal side of one angle is the initial side of the other.
Arcs and Sectors
The radius of the circle is one of the factors that determines the length s of the arc. Therefore, the radian measure θ of an angle equals the length of the arc s/radius r. Similarly, the area of a circular sector equals ½ r2θ. These values are only true when the angle θ is measured in radians, as those measurements are standardized for the radius of a circle.
Circular Motion
Suppose a point is moving around the outside of a circle, for example, if an object is in a circular orbit in space. There are two ways to describe the rate that it is moving. The linear speed is the distance traveled divided by the elapsed time. The angular speed is the change in the measure of the central angle in radians divided by the elapsed time. In math language, the linear speed v equals the distance s/time t, or v=s/t. The angular speed is represented by the Greek letter omega ω, which equals the angle θ/time t, or ω =θ/t. Linear and angular speed are related such that linear speed v =radius r times angular speed ω, or v= rω. This is a very useful concept in circular mechanics and in orbital motion.
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