Objectives
• Change from radian to degree measure,
and vice versa
• Find angles that are co-terminal with a
given angle
• Find the reference angle for a given
angle
Angles and Their Measures
• Angles are formed by the rotation of two rays that
share a common and fixed endpoint.
• If one ray remains fixed, it forms the initial side.
• The second ray rotates to form the terminal side.
• If the rotation is in a counter-clockwise direction, it
forms a positive angle.
• If the rotation is in a clockwise direction, it forms a
negative angle.
• An angle that has its vertex at the origin and its
initial side along the x-axis, the angle is said to be in
standard position.
The angles below are all in standard position
Initial side
Terminal side
Positive angle
Negative angle
Terminal side
Terminal Side
Initial side
Initial Side
Quadrantal Angle
Angle Measurement
• There are two common units used to
measure angles, degrees and radians.
• An angle is 1° if it is 1/360 of a revolution
in the positive direction.
• Each degree consists of 60 minutes – 60’.
• Each minute consists of 60 seconds –
60”.
Change 29°45’26” to a decimal
number of degrees to the nearest
one thousandth
• 29° + 45’(1/60) + 26”(1/3600)
= 29° + .750° + .007°
= 29.757°
Radians
• Another unit of measure is the radian.
• The definition of a radian is based on the unit
circle. A unit circle is a circle with a radius of
1.
• There is an important relationship between
degrees and radians.
• Since one revolution of a circle is either 360°
or 2p radians:
360° = 2p radians 180° = p radians
90° = p/2 radians
45° = p/4 radians
Degree/Radian Conversion Formulas
• To change degrees to radians:
multiply by p/180 radians
• To change radians to degrees:
multiply by 180°/p
• 1 radian = about 57.3°
• 1 degree = about 1.017 radians
Tomorrow - co-terminal angles and reference angles
Assignment
• Make a unit circle with multiples of 30°
– Label both degree and radian measures
• Make a unit circle with multiples of 45°
-- label both degree and radian measures