Angles and their Measures

Convert between degrees minutes seconds and decimal forms of angles

Find the arc lengthConvert from degrees to radians

Find the area of a sector of a circleFind the linear speed of an object traveling in circular motion

Angles

A ray is that portion of a line that starts at a point on the line and extends indefinitely in on direction

The starting point of the line is called the vertexIf two rays are drawn with a common vertex

they form an angleWe call one of the rays of an angle the initial

side and the other the terminal side

Continued

• The angle that is formed is identified by showing the direction and amount of rotation from the initial side to the terminal side.

• A rotation that counterclockwise is positive• A rotation that is clockwise is negativeWe use lower case Greek letters to identify

these angles

An angle θ is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with the positive x axis.

When an angle is in standard position, the terminal side will lie either in quadrant in which case we say that the angle lies in that quadrant

If the angle lies on the x axis or y axis then we say that the angle is a quadrantal angle

Smaller than one degree

Subdivisions of a degree can be expressed by using a decimal we also will use notation minutes and seconds

One minute is defined as of a degree and one second is

60

1

60

1

Converting between Degrees minutes seconds and decimals

Convert 50°6’21”

Convert 21.256 to the DMS form

Central Angles

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle intersects an arc on the circle

Radians

If the radius of the circle is r and the length of the arc subtended by the central angle is also r, then the measure of the angle is 1 radian.

For a circle of radius 1 the rays of a central angle with measure 1 radian would intersect an arc length 1

Relationship between Degrees and Radians

1 revolution = 2Π

180°= Π radians

1 degree= radians

1 radian = degrees

180

180

Examples

Convert 60° to radians

Convert to degrees 4

3

Commonly encountered anglesDegrees 0 30 45 60 90 120 135 150 180

Radians

Degrees 210 225 240 270 300 315 330 360

Arc length

For a circle of radius r a central angle of θ radians subtends an arc whose length s is

s=rθ

ONLY RADIANS

Degrees

Area of a sector

The area A of a sector of a circle of radius r formed by a central angle θ radians is

A = r2θ2

1

Circular Motion Linear Speed

Last quarter: The average speed of an object as the distance traveled divided by the elapsed time

Suppose an object travels around a circle of radius r at a constant speed. If s is the distance traveled in time t around this circle then the LINEAR SPEED

t

s

Circular Motion Angular Speed

As this object travels around the circle suppose that θ (only measured in radians) is the central angle swept out of time t.

The angular speed ω (omega) of this object is the angel swept out divided by the elapsed time

t