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4.1 Angles and their Measures

What is an angle and how do we measure it?

An angle is a measurement of a rotation or change of direction. 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 unit of rotation is the revolution. If an object makes one revolution, it turns one time about some axis to return to its original orientation. If it turns 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 \(300\) rpm, one is saying that the wheel turns through an angle of \(300\) revolutions per 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. We can thank the ancient Babylonians for the degree measure of angles as well as the minutes and seconds we use to measure time. Whereas we use a base \(10\) number system, the ancient Babylonians used base \(60\). So when dividing up the circle they first divided it up into six equal parts using a common geometric construction, then subdivided that one-sixth part of a circle into \(60\) smaller parts to get the degree unit for measuring angles.

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. This unit of measure is slightly smaller than the one-sixth of a circle used by the Babylonians, since it is about \(\dfrac{1}{6.2832}\) part of a circle, \(\dfrac{1}{2\pi}\) to be exact. Because a point moving a full circle about the circumference of a circle will travel exactly a distance of \(2\pi\) times the length of a radius of the circle. Though this may seem a strange way to measure a rotation, its use greatly simplifies the derivative rules of trigonometric functions in the study of calculus. In the study of calculus, radian measure of angles is preferred.

You may hear it said that a radian is a unitless measure. This statement is true in one sense of the term 'unit' but false in another sense. It is true in the sense that it is defined as a ratio between two distances, and the units of measure of the two distances 'cancel out' leaving no units. And in formulas from physics, for example, rates of rotation are measured in units of 'per seconds', meaning 'radians per second.' But in another sense of the term 'unit' the statement that radians are unitless is false. Every quantity that is measurable is measured in some unit of that quantity that serves as a standard of measurement. For example, length may be measured in meters, with one meter being the 'unit' of length. Similarly, angles can be measured in terms of the unit 'radians,' which is a specific angle which serves as a standard unit for measuring angles.

One revolution equals \(360^\circ\) equals \(2\pi\) radians.

Exercise 4.1.1

Convert \(45^\circ\) to revolutions and to radians.

See Solution

Minutes and seconds versus decimal fractions of a degree

Historically, fractions of a degree have been measured using the Babylonian units of minutes and seconds. Since the introduction of pocket calculators, however, this method of measuring fractions of a degree has gradually been replaced with decimal fractions of a degree.

Many calculators have a special key to convert between minutes and seconds to decimal fractions of a degree and visa versa, but the calculations are simple to do in the absence of such a key.

For example, let us convert \(32^\circ 42^\prime 14^{\prime\prime}\) to degrees and decimal fractions of a degree. We begin at the small end with the seconds and work backwards, dividing by \(60\) at each step and rounding the result at the third place past the decimal: \([(14\div60+42)\div60+32]=32.704^\circ\).

To convert from decimal fractions of a degree to minutes and seconds, multiply the decimal part by \(60\) and take the whole number part of the result as the number of minutes. Take the fractional part of the result and multiply by \(60\) and round to the nearest whole number to get the seconds.

For example, convert \(78.314^\circ\) to degrees, minutes and seconds: \(0.314^\circ\times60=18.84^\prime\), \(0.84^\prime\times60=50.4^{\prime\prime}\). So \(78.314^\circ=78^\circ18^\prime50^{\prime\prime}\).

Exercise 4.1.2

Convert \(32^\prime 14^{\prime\prime}\) to decimal fractions of a degree.

See Solution

Exercise 4.1.3

Convert \(42.57^\circ\) to degrees, minutes and seconds.

See Solution

Converting between degrees and radians

To convert from degrees to radians, multiply by the conversion factor \(\dfrac{\pi}{180^\circ}\).

Example: \(75^\circ\times\dfrac{\pi}{180^\circ}=\dfrac{5\pi}{12}\) radians. When the result is a simple fraction of or multiple of \(\pi\) the result is usually left in that form. Otherwise it is converted to decimal radians.

Convert \(38.629^\circ\) to radians: \(38.629^\circ\times\dfrac{\pi}{180^\circ}=0.6742\) radians. As a rule, when converting to radians, include one additional decimal place of accuracy.

Converting from radians to degrees, multiple by the conversion factor \(\dfrac{180^\circ}{\pi}\).

For example, convert \(\dfrac{8\pi}{15}\) radians to degrees: \(\dfrac{8\pi}{15}\times\dfrac{180^\circ}{\pi}=96^\circ\).

Convert \(1.2143\) radians to degrees: \(1.2143\times\dfrac{180^\circ}{\pi}=69.574^\circ\).

Exercise 4.1.4

Convert 1 radian to revolutions and to degrees.

See Solution

Exercise 4.1.5

Convert \(\dfrac{5\pi}{12}\) radians to revolutions and to degrees.

See Solution