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Module 5-I – Introduction to Trigonometry

Section 5-I.2 – Measurement of Angles

 

 

Introduction

This section reviews various ways of measuring angles. We introduce and emphasise a unit of measurement which you may not have seen before (the radian), but which is convenient in situations where we want simple relations between lengths and angles.

 

 

Section Learning Objectives

After completing this unit you should be able to:

·        Define the degree, minute, second and radian units of angular measurement

·        Convert  the description of an angle from one unit to another

·        Quickly estimate the measure of a given angle in any of these units

·        Solve geometric and applied problems involving the determination of arc lengths in terms of angles

 

 

 

What to Do

1.         Read the following Study Notes and Discussion

2.         Read section 5.1 of the text.

4.         Follow the instructions regarding Further Practice.

Study Notes and Discussion

 

By the size of an angle we mean the amount of turning needed to rotate one arm to the other (independent of the lengths of the arms). So a natural way to measure angles is in terms of a full revolution. But then of course most of the angles we draw would be small fractions of a revolution, so it is natural to choose some small fraction of a revolution as the basic unit of angle. 

 

Degrees, Minutes, and Seconds of arc

Historically, the fraction chosen was 1/360 and the resulting unit is the familiar 'degree'. (One reason for this rather odd looking choice might have been that 360 has a lot of factors so it is easy to express a lot of simple fractions of a revolution as a whole number of degrees. Another possible reason is that 360 is close to the number of days in a year and so one degree is close to the angle by which the sun appears to progress each day relative to the distant stars.)  Angles smaller than a degree are often expressed in decimal form, but sometimes also in terms of the smaller units of minute, 1' = (1/60)°, and second, 1" = (1/60)'.

 

What is the angle covered in a minute by the minute hand of a clock?(see answer #1)

 

 

Radian Measure of Angles

But degrees, minutes, and seconds are not the only common units of angular measure. 

 

Another unit which is especially suited to relating angles and lengths or distances is the so-called 'radian'(1^r) which is the angle subtended at the centre of a circle by an arc of length equal to its radius.

 

The radian measure of an angle is thus equal to the ratio of arc length to radius (which, by similarity, is independent of the actual lengths or the unit in which they are measured).  So conversely, the length of a circular arc is just equal to the length of the radius multiplied by the radian measure of the angle at the centre, and for a radius of 1 unit, the radian measure of an angle is exactly equal to the arc length

 

            Since the circumference of a circle is 2π times its radius,

we see at once that

                                    360°  =  1rev.  =  2π^r

and so                          1° = (1/360)rev. =(2π/360)^r

and                               1^r = (1/2π)rev.  =(360/2π)°

Since  is just a bit more than 6, one radian is just under  of a revolution, or a bit less than 60°. (In fact it’s about 57.3°)

           

How long does it take the minute hand of a clock to turn through an angle of one radian?

(see answer #2)

 

 

 

Further Practice

Work through some of  the odd numbered questions on pages369-371 (using the answers and solutions manual to check your work).Do enough to convince you that you can do all types easily. As a bare minimum, do ##9,19,31,45, and 51.