4.1  Angles and their Measures

 

An angle is a measurement of a rotation.  In plane geometry, an angle is the measurement of the rotation required to move one ray onto another ray emanating from the same point.  A ray is a ‘half-line’ beginning at some point P and extending indefinitely in one direction.  By convention, counterclockwise rotations have positive measurements and clockwise rotations have negative measurements.  An exception is that in plane geometry, sometimes all angles are considered to be positive, regardless of the direction of rotation.

 

When measuring rotations, various unit rotations may be used.  A unit is a reference quantity with respect to which all other quantities may compared in order to find their measure.

 

One rotational unit is the revolution.  If an object makes one revolution, it turns one time about some axis to return to its original orientation.  If it makes one-half revolution, it will end up oriented opposite to its original orientation.  When one says that a wheel is turning at a rate of 300 rpms, one is saying that the wheel turns through an angle of 300 revolutions each minute.

 

Another traditional unit of rotation is the degree.  The degree is one three hundred sixtieth of one revolution.  A degree may be divided into 60 smaller units called minutes and the minute may be subdivided further into 60 smaller units called seconds.

 

A modern unit of rotation is the radian.  When a circle of radius r rotates by some amount, a point on its circumference moves some distance along the circumference.  The ratio of that distance to the radius of the circle is the radian measure of the rotation.  The angle is one radian if the point moves a distance along the circumference equal to the radius of the circle.

 

Degree measure is traditionally used in geometry, revolution measure is traditionally used when describing large rotation rates, and radian measure is traditionally used in calculus.  One revolution equals 360o, and  radians.

 

Exercise 4.1.1

 

Convert 45 o to revolutions and to radians.

 

Solution

 

Exercise 4.1.2

 

Convert 32’ 14” to decimal fractions of a degree.

 

Solution

 

Exercise 4.1.3

 

Convert 42.57 o to degrees, minutes and seconds.

 

Solution

 

Exercise 4.1.4

 

Convert 1 radian to revolutions and to degrees.

 

Solution

 

Exercise 4.1.5

 

Convert  radians to revolutions and to degrees.

 

Solution

 

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