Radian and Degree Measure

Objectives:Describe Angles

Use Radian and Degree measures

An angle is determined by rotating a ray about its endpoint. The starting position of the ray is the initial side of the angle and the position after rotation is the terminal side. The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x-axis. Such an angle is in standard position. Counterclockwise rotation generates positive angles and clockwise rotation generates negative angles. Angle that have the same initial and terminal sides are called coterminal angles.

Trigonometry: the measurement of trianglesthe relationships among the sides and angles of

triangles

Radian: the measure of a central angle that intercepts an arc equal in length to the radius of the circle. Algebraically, this means that where is measured in radians.

(Note that )

Degree: a measure of one degree is equivalent to a rotation of of a complete revolution about the vertex.

Measure of an angle: the amount of rotation from the initial side to the terminal side.

s rs

r

1 when s r

1

360

Conversions between Degrees and Radians

To convert degrees to radians, multiply degrees by

EX:Convert from degrees to radiansa)

b)

To convert radians to degrees, multiply radians by

EX:Convert from radians to degreesa)

b)

180

rad

180

rad

135

5402

7

6

Fractional parts of degrees are expressed in minutes and seconds, using the prime and double prime notations, respectively. Many calculators have special keys for converting an angle in degrees, minutes, and seconds to decimal degree form, and vice versa.

Decimal degrees are used to denote fractional parts of degrees

11 one minute = 1

601 1

1 one second 1 160 3600

A) Determine the quadrant in which the angle lies B) Convert to degrees C) Determine two coterminal angles (one positive

and one negative) for each angle. D) Determine the complement and the supplement

of each angle (IF POSSIBLE).

1.

EXAMPLE

2

3

A) Determine the quadrant in which the angle lies B) Convert to radians C) Determine two coterminal angles (one positive

and one negative) for each angle. D) Determine the complement and the supplement

of each angle (IF POSSIBLE).

2.

EXAMPLE

75

A) Determine the quadrant in which the angle lies B) Convert to degrees C) Determine two coterminal angles (one positive

and one negative) for each angle.

3.

EXAMPLE

9

4

A) Determine the quadrant in which the angle lies B) Convert to radians C) Determine two coterminal angles (one positive

and one negative) for each angle.

4.

EXAMPLE

290

a)

b)

c)

EX 5: Convert to Decimal Degrees. Round to the nearest thousandth of a degree.

74 28 15

135 36

54 45

a)

b)

c)

EX 6: Convert to Degree, Minutes and SecondsD M S

240.6

34.817

345.12