Section 5.1 Angles and Arcs Objectives of this Section Convert Between Degrees, Minutes, Seconds, and Decimal Forms for Angles Find the Arc Length of a Circle Convert From Degrees to Radians, Radians to Degrees Find the Linear Speed of Objects in Circular Motion

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. VRay

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. Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle

Vertex Initial Side Terminal side Clockwise rotation Negative Angle Vertex Initial Side Terminal side Counterclockwise rotation Positive Angle

Initial sideVertex Terminal side x y

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. x y x y

Angles are commonly measured in either Degrees or 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 Initial side Terminal side Vertex

Initial side Terminal side Vertex

Initial sideTerminal sideVertex

x y Initial sideVertex Terminal side

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

For a circle of radius r, a central angle of radians subtends an arc whose length s is Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 2 radians.

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

Let (measured in radians) be the 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.

To find relation between angular speed and linear speed, consider the following derivation.

Acknowledgement Thanks to Addison Wesley and Prentice Hall. These notes are taken from Sullivan Algebra and Trigonometry