Download Angles

An angle in standard position has its vertex at the origin and its initial side on the positive x-axis.
– Rotating counterclockwise yields a positive angle.
– Rotating clockwise yields a negative angle.
Coterminal angles are angles with the same initial and terminal sides.
A central angle is an angle whose vertex is at the center of a circle.
An angle θ is acute if 0o < θ < 90o.
An angle θ is obtuse if 90o < θ < 180o.
Two angles α and β are complementary if α + β = 90o.
Two angles α and β are supplementary if α + β = 180o.
Measuring Angles
A degree, 1o, is 1/360 of a revolution, circle.
– A minute, 1’, is 1/60th of a degree.
– A second, 1”, is 1/60th of a minute or 1/3600th of a degree.
A radian is the measure of a central angle that intersects an arc of length equal to the radius.
– Since the circumference of a circle is 2πr, a circle has an angle measure of 2π radians.
– What is the angle measure of a semicircle, quarter circle, a third of a circle, two revolutions?
– Radians have no units of measurement.
Conversions
To convert from radians to degrees, multiply the angle by 180o/π.
To convert from degrees to radians, multiply the angle by π/180o.
Examples
74o
105o
-193o
740o
182.37o
57o41’12”
5π/6
π/4
2π/7
-11π/9
3
-13.23
Applications
An arc is the portion of a circle between the initial and terminal sides of a central angle.
The length of the arc or arc length, s, is given by s = rθ where r is the radius of the circle and θ is
the angle measure in radians.
A sector is the region bounded by the initial and terminal sides of a central angle and the arc
between them.
The area of a sector, A, is given by A 

2
r 2 where r is the radius of the circle and θ is the angle
measure in radians.
Examples
A central angle of 2 is in a circle with radius 5 cm.
a. Find the arc length of the arc created by the central angle.
b. Find the area of the sector created by the central angle.
A central angle of 73o is in a circle with radius 2 ft.
a. Find the arc length of the arc created by the central angle.
b. Find the area of the sector created by the central angle.
Let an object be traveling in a circle.
Speed is the change in position over the change in time.
 Usually, the change in position is given as a distance.
 r = d/t
 This is the linear speed.
 However, the change in position can be given as an angle.
 This is the angular speed.
The linear speed of the object, v, is given by v = s/t where s is the arc length.
The angular speed of the object, ω, is given by ω = θ/t where θ is the angle in radians.
Examples
The minute hand on a clock is 6 in. long.
a. Find the linear speed of the tip of the minute hand as it moves from the 2 to the 8.
b. Find the angular speed of the tip of the minute hand as it moves from the 2 to the 8.
c. Find the linear speed of the tip of the minute hand as it moves 37 minutes.
d. Find the angular speed of the tip of the minute hand as it moves 37 minutes.
A 20 in. diameter bike tire with a red dot is spinning at 15 revolutions per minute.
a. Find the linear speed of the red dot.
b. Find the angular speed of the red dot.
c. Suppose the diameter is 24 in. Find the linear and angular speeds of the red dot.
A car is going 55 mph.
a. Suppose the tire diameter is 25 inches. Find the angular speed.
b. Suppose the tire diameter is 30 inches. Find the angular speed.