Sullivan Algebra and Trigonometry: Section 6.1 Angles and Their Measures Objectives of this Section Convert Between Degrees, Minutes, Seconds, and Decimal Forms for Angles Find the Arc Length of a Circle Convert From Degrees to Radians, Radians to Degrees Find the Area of a Sector of a Circle Find the Linear Speed of Objects in Circular Motion A ray, or half-line, is that portion of a line that starts at a point V on the line and extends indefinitely in one direction. The starting point V of a ray is called its vertex. V
Ray If two lines are drawn with a common vertex, they form an angle. One of the rays of an angle is called the initial side and the other the terminal side. Ter n i m e d i al s Vertex
Initial Side Counterclockwise rotation Positive Angle e d i s al n i Term Vertex Initial Side
Clockwise rotation Vertex Ter Negative Angle e d i s l mina Initial Side
Counterclockwise rotation Positive Angle An angle is said to be in standard position if its vertex is at the origin of a rectangular coordinate system and its initial side coincides with with positive x - axis. Terminal side y Vertex Initial side x When an angle is in standard position, the
terminal side either will lie in a quadrant, in which case we say lies in that quadrant, or it will lie on the x-axis or the y-axis, in which case we say is a quadrantal angle. y y x is a quadrantal angle lies in Quadrant III x Angles are commonly measured in either Degrees or Radians
The angle formed by rotating the initial side exactly once in the counterclockwise direction until it coincides with itself (1 revolution) is said to measure 360 degrees, abbreviated 360 . Terminal side Initial side Vertex 1 One degree, 1 , is revolution. 360 1 A right angle is an angle of 90 , or 4 revolution.
Terminal side Vertex Initial side 1 90 angle; revolution 4 A straight angle is an angle of 180 , 1 or revolution. 2
Terminal side Vertex Initial side 1 180 angle; revolution 2 -135 Drawaa-135 angle. angle. Draw y
Vertex Initial side Te r m in al s id e x
135 One minute, denoted, 1, is defined as 1 degree. 60 One second, denoted, 1, is defined as 1 1 second, or degree. 60 3600 1 counterclockwise revolution = 360
60 = 1 60 = 1 Convert 30 1255 to a decimal in degrees. 1 1 30 1255 30 12 55 60 3600
30 0.2 0.015278 30.215278 Convert 45.413 to D M S form. 60 0.413 0.413 24.78 1 60
0.78 0.78 46.8 47 1 45.413 45 24 47 Consider a circle of radius r. Construct an angle whose vertex is at the center of this circle, called the central angle, and whose rays subtend an arc on the circle whose length is r. The measure of such an angle is 1 radian.
r r 1 radian For a circle of radius r, a central angle of radians subtends an arc whose length s is s r Find the length of the arc of a circle of radius 4 meters subtended by a central angle of 2 radians. r 4 meters and = 2 radians s r 4 2 8 meters
1 revolution = 2 radians 180 radians 1 degree = radian 180 180 1 radian = degrees Convert 135 to radians. radian 135 135
180 3 radian 4 2 Convert radians to degrees. 3 2 2 180 radians = 3
3 120 The area A of the sector of a circle of radius r formed by a central angle measured in radians is 1 2 A r 2 Example: Find the area of the sector of the circle of radius 4 inches formed by an angle of 45 degrees. 45 45 180 4
rad 1 2 A (4 ) 2 4 A 2 6.28 in 2 Suppose an object moves along a circle of radius r at a constant speed. If s is the distance traveled in time t along this circle, then the linear speed v of the object is defined as s v
t As the object travels along the circle, suppose that (measured in radians) is the central angle swept out in time t. Then the angular speed of this object is the angle (measured in radians) swept out divided by the elapsed time. t To relate angular speed with linear speed, consider the following derivation. s r s r
t t v r A cars engine is running at 2000 rpm. If the length of each crank in the engine is 12 inches, find the linear speed of the piston. 2000 revolutions 2 radians minute revolution 4000 radians / minute 4000 v r 12 inches minute 150,796 inches / minute