Angles and Radian Measure

4.1 – Angles and Radian Measure

An angle is formed by rotating a ray around its endpoint. The original position of the ray is called the initial side of the angle, and the ending position of the ray is called the terminal side. The endpoint of the ray

is called the vertex of the angle.

vertex

terminal side

initial side

4.1 – Angles and Radian Measure

An angle is said to be in standard position if its vertex is at the origin, and its initial side is on the positive x-axis.

An angle is positive if it is rotated counterclockwise, negative if it is rotated clockwise.

positive

negative

4.1 – Angles and Radian Measure

Degree Measure

90º (right angle)

180º (straight angle)

270º

360º

4.1 – Angles and Radian Measure

Degree Measure

90º

180º 360º

270º

acute(between 0º and 90º)

obtuse(between 90º and 180º)

Converting from Degrees-Minutes-Seconds (DMS) to Decimal Degrees (DD)

“DMS” stands for Degrees-Minutes-Seconds.

The conversions are based on subdivisions of a whole degree where…1 degree = 60 minutes1 minute = 60 seconds

It follows that 1 degree = 3600 seconds

From DMS to DD

360060"'

SMDSMD

From DD to DMS1. Multiply the decimal part of the

degrees (to the right of the decimal point) by 60. These are your minutes.

2. Now multiply the decimal part of your minutes (to the right of the decimal point) by 60. These are your seconds.

*There is a DMS button on your calculator.*

Examples

35 degrees 23 minutes 57 seconds to DD

(round to four decimal places)

97.554 degrees to DMS

35 23'57"

4.1 – Angles and Radian Measure

If θ is an angle with its vertex at the center of a circle of radius r, and θ intercepts an arc of length s on that circle,

then θ = radians.

s

θ r

sr

r

4.1 – Angles and Radian Measure

If s is the length of an arc intercepted by a central

angle θ (in radians) in a circle of radius r, then s = rθ.

s

θ r

4.1 – Angles and Radian Measure

If θ intercepts an arc of length equal to the radius of the circle, θ has a measure of 1 radian.

r

θ r

4.1 – Angles and Radian Measure

One complete rotation

s θ r

s = circumference = 2πr

θ = 2πr

r = 2π radians

4.1 – Angles and Radian Measure

Radian Measure

(right angle)

π (straight angle)

2π

2π

23π

180 =

4.1 – Angles and Radian Measure

To convert from degrees to radians, multiply by .

To convert from radians to degrees, multiply by .

60º =

5 radians

degrees 180

radians π60 degrees

3

180

π

π

180

radians3

π

radians π

degrees 180= 150º

30

radians6

5π

radians π

degrees 180= 286.48º

ExamplesConvert the angle in degrees to radians.

Convert the angle in radians to degrees.

) 18 ) 250a b

11 5) )6 2

a b

4.1 – Angles and Radian Measure

Two angles are said to be coterminal if they have the same terminal side.

40º

Angle = 40º

Coterminal angle: 40º − 360º = −320º

40º + 360º = 400º

40º + 720º = 760º

4.1 – Angles and Radian Measure

Two angles are said to be coterminal if they have the same terminal side.

Angle =

Coterminal angle:

6π

6π

π26π 6

11π

π26π 6

13π

π46π 6

25π

4.1 – Angles and Radian Measure

A clock has a minute hand which is 8 inches long. If the minute hand moves from 12:00 to 3:00, how far

does the tip of the minute hand move?

s = rθ

s = rθ

= (8)(90) = 720 inches

2π(8) = 4π = 12.57 inches

4.1 – Angles and Radian Measure

Two angles whose sum is 90º are called complementary.

Two angles whose sum is 180º are called supplementary.

40º

Angle = 40º

Complement: 90º − 40º = 50º

Supplement: 180º − 40º = 140º