5.1 Angles and Their Measure

A ray, or half-line, is that portion of a line that starts at a point V on the line and extends indefinitely in one direction. The starting point V of a ray is called its vertex. Ray V

If two lines are drawn with a common vertex, they form an angle If two lines are drawn with a common vertex, they form an angle. One of the rays of an angle is called the initial side and the other the terminal side.

Terminal side Initial Side Vertex Counterclockwise rotation Positive Angle

Terminal side Vertex Initial Side Clockwise rotation Negative Angle

Terminal side Vertex Initial Side Counterclockwise rotation Positive Angle

y Terminal side x Vertex Initial side

When an angle is in standard position, the terminal side either will lie in a quadrant, in which case we say lies in that quadrant, or it will lie on the x-axis or the y-axis, in which case we say is a quadrantal angle.

y x

y x

Angles are commonly measured in either 1) Degrees 2) Radians

The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated Terminal side Initial side Vertex

Terminal side Vertex Initial side

Terminal side Initial side Vertex

y Vertex Initial side x Terminal side

Consider a circle of radius r Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.

r 1 radian

Theorem Arc Length For a circle of radius r, a central angle of radians subtends an arc whose length s is

r = 4 meters and = 2 radians = 4(2) = 8 meters. Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 2 radians. r = 4 meters and = 2 radians = 4(2) = 8 meters.

Suppose an object moves along a circle of radius r at a constant speed Suppose an object moves along a circle of radius r at a constant speed. If s is the distance traveled in time t along this circle, then the linear speed v of the object is defined as

As the object travels along the circle, suppose that (measured in radians) is the central angle swept out in time t. Then the angular speed of this object is the angle (measured in radians) swept out divided by the elapsed time.