angles and their measures convert between degrees minutes seconds and decimal forms of angles find...
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Angles and their Measures
Convert between degrees minutes seconds and decimal forms of angles
Find the arc lengthConvert from degrees to radians
Find the area of a sector of a circleFind the linear speed of an object traveling in circular motion
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Angles
A ray is that portion of a line that starts at a point on the line and extends indefinitely in on direction
The starting point of the line is called the vertexIf two rays are drawn with a common vertex
they form an angleWe call one of the rays of an angle the initial
side and the other the terminal side
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Continued
• The angle that is formed is identified by showing the direction and amount of rotation from the initial side to the terminal side.
• A rotation that counterclockwise is positive• A rotation that is clockwise is negativeWe use lower case Greek letters to identify
these angles
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An angle θ 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 the positive x axis.
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When an angle is in standard position, the terminal side will lie either in quadrant in which case we say that the angle lies in that quadrant
If the angle lies on the x axis or y axis then we say that the angle is a quadrantal angle
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Smaller than one degree
Subdivisions of a degree can be expressed by using a decimal we also will use notation minutes and seconds
One minute is defined as of a degree and one second is
60
1
60
1
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Converting between Degrees minutes seconds and decimals
Convert 50°6’21”
Convert 21.256 to the DMS form
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Central Angles
A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle intersects an arc on the circle
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Radians
If the radius of the circle is r and the length of the arc subtended by the central angle is also r, then the measure of the angle is 1 radian.
For a circle of radius 1 the rays of a central angle with measure 1 radian would intersect an arc length 1
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Relationship between Degrees and Radians
1 revolution = 2Π
180°= Π radians
1 degree= radians
1 radian = degrees
180
180
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Examples
Convert 60° to radians
Convert to degrees 4
3
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Commonly encountered anglesDegrees 0 30 45 60 90 120 135 150 180
Radians
Degrees 210 225 240 270 300 315 330 360
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Arc length
For a circle of radius r a central angle of θ radians subtends an arc whose length s is
s=rθ
ONLY RADIANS
Degrees
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Area of a sector
The area A of a sector of a circle of radius r formed by a central angle θ radians is
A = r2θ2
1
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Circular Motion Linear Speed
Last quarter: The average speed of an object as the distance traveled divided by the elapsed time
Suppose an object travels around a circle of radius r at a constant speed. If s is the distance traveled in time t around this circle then the LINEAR SPEED
t
s
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Circular Motion Angular Speed
As this object travels around the circle suppose that θ (only measured in radians) is the central angle swept out of time t.
The angular speed ω (omega) of this object is the angel swept out divided by the elapsed time
t