4.1 Radian and Degree Measure

Objective

• To use degree and radian measure.

Angles

• An angle is determined by rotating a ray about its endpoint.

• The starting point of the ray is the initial side.

• The ending position is the terminal side.

Angles

• The endpoint of the ray is called the vertex of the angle.

• An angle in standard position has its initial side on the positive x-axis and its vertex at the origin.

Angles

• If the rotation of the ray is counterclockwise the angle has positive measure.

• If the rotation of the ray is clockwise the angle has negative measure.

Angles

• An angle in standard position is said to lie in the quadrant in which its terminal side lies.

• A central angle of a circle is an angle whose vertex is on the center of the circle.

Degree Measure

• An angle generated by one complete counterclockwise rotation measures 360°.

• One generated by a complete clockwise rotation measures -360°.

Coterminal angles

• Angles that have the same initial and terminal sides are called coterminal.

Radian Measure

• The measure of an angle is determined by the amount of rotation from the initial side to the terminal side.

Definition of a Radian

• One radian is the measure of a central angle θ that intercepts an arc s equal in length to the radius r of the circle. Algebraically this means that θ = s / r where θ is measured in radians.

Radian Demo http://id.mind.net/~zona/mmts/trigonometryRealms/radianDemo1/RadianDemo1.html

• The circumference of a circle is 2πr. There are 2π radians around the circumference of a circle.

• There are about 6.28 radians in a full circle.

• 1 radians = 57.29°

• 1 degree = 0.017 radians

• To convert degrees to radians:

• To convert radians to degrees

• When no units of angle measure are specified, radian measure is implied.

ExampleConverting from Degrees to

Radians

• 135° 3π/4

• 540° 3π

• -270° -3π/2

ExampleConverting from Radians to

Degrees

902

2 114.6

9810

2

rad

rad

rad