Angles - The UEA Portal

Angles - The UEA Portal
Learning Enhancement Team
Steps into Trigonometry
Angles
This guide introduces the definition of different types of angles and the
different ways of measuring them. It also gives formulas to covert
between degrees and radians.
Introduction
Angles are fundamental to geometry and having an understanding of angles can help
you expand the way you look at mathematics. Making the connection between
geometry and (the more commonly taught) algebra is important and often overlooked.
Much of early mathematics was concerned with shapes and the ancient Greeks certainly
knew about angles. However the most famous Greek treatise on mathematics, Euclid’s
Elements only describes one angle (a right-angle) and describes all other angles as
parts or multiples of a right angle. When astronomy became a more formalised pursuit,
leading to the advent of trigonometry, angles naturally played an important role.
Through Hipparchus and eventually Ptolemy, triangles could be solved (both plane and
spherical) and the angle was firmly placed at the heart of geometry. In your studies you
will definitely come across angles in trigonometry and they are essential to
understanding the trigonometric ratios (see study guides: Trigonometric Ratios: Sine,
Cosine and Tangent, Solving Right-Angled Triangles, Further Trigonometry, The Sine
Rule and The Cosine Rule). You will also find they play a role in calculus and vectors
and so in science in general.
An angle is made when two straight lines cross. The lines AB and BC are called the
sides of the angle and the point where they meet, B, is called the vertex of the angle.
It is usual to draw an arc joining the two lines to depict the angle. In the diagram below
you can see a drawing of an angle:
A

B
C
Here A, B and C refer to the three points that define the angle which is labelled by the
symbol . You will find that is very common to name angles using italic Greek letters
such as , ,  and so on. There are many other different ways to denote an angle, you
might commonly encounter the angle  being described by ABC, AB̂C , B̂ or ABC .
Units used to measure angles
As in any physical quantity, there are different unit systems to measure angles. In
mathematics and science the degree and the radian are the most widely used. Many
mistakes in the calculation of angles occur because your calculator may be set in the
wrong mode. For example, you may have your calculator in degree mode when you
need an answer in radians. You should always check if your calculator is in the
appropriate mode for a calculation. You can consult the manual for your calculator to
help you understand how to change the mode on your calculator. Most calculators have
three modes to perform calculations involving angles degrees, radians and gradians:
a)
Degrees. Symbol , 1  0.017 rad  1.11 grad. In this unit system the full rotation
around a circle is divided into 360 equal parts. Each part called a degree (from
the Latin de gradus) and would be written 1. The origin of this system is not clear
with some assigning a link to Babylonian culture whose year had 360 days (plus
five bad luck days). Others suggest it is based on the base 60 numeric system
(sexagesimal) as 60 is a number which can be divided by many smaller whole
numbers. One degree can be subdivided into 60 equal parts, each named a
minute, which is written 1 . In turn, each minute can be subdivided in 60 equal
parts, each one called a second, which is written 1 . Most angles are written in
decimal form such as 47.5. However in navigation bearings, angles are written in
terms of degrees, minutes and seconds, for example 47 31 13 and are said 47
degrees 31 minutes and 13 seconds.
b)
Radians. Symbol rad or no symbol, 1 rad  57.3  63.66 grad. One radian (or
radius angle) is the angle centred on a circle whose arc is the same length as the
radius of that circle. Therefore the length of the arc of a full rotation in a circle is 2
radians (see table below), you may know that the circumference of a circle is 2
multiplied by the radius. It is very common to express radians as multiples of .
Although the concept of measuring angles in terms of arc length has been around
for a long time, the term ‘radian’ was only first used by James Thomson in 1871.
The use of radians as a way of measuring angles is very important when working
with calculus. All calculus calculations involving a trigonometrical element are
performed in radians. There are also some formulas which only work when the
angle concerned is expressed in radians. For example equations for finding the
arc length and area of a sector in a circle. In fact the radian is the SI unit for angle.
c)
Gradian. Symbol grad, 1 grad  0.9  0.016 rad. A gradian is defined so that a
full rotation is 400 grad and so a right angle is 100 grad. They were introduced to
try and decimalise the measurement of angles. This system is rarely used today
but you will find it as an option on most calculators.
Classification of angles
There are different types of angles, and they can be classified according to size. These
are: acute, right, obtuse, straight and reflex angles as well as a full rotation or turn.
In the table on the next page you can find an example for each case and how they are
defined in terms of both degrees and radians.
Type of Angle
Figure
Range (degrees)
Range
(radians)

Acute
0    90
0  
Right
  90

Obtuse
90    180
Straight
  180
 
Reflex
180    360
    2
Full rotation or
Turn
  360
  2

2
2

2
 
You are often required to convert an angle from degrees to radians or from radians to
degrees. You can try to remember the formulas but it is better to understand their origin,
this will allow you to derive the relationships quickly. You start by recognising that all the
way around a circle can be described by either 360 or 2 radians. This implies that the
two are equal.
360  2 radians
If you divide both sides of this relationship by 360 then you get a relationship for 1:
1 

180
radians
You can then work out a relationship for any number of degrees (say n) by multiplying
both sides by n:
n 
n
radians
180
Similarly, if you divide both sides of 360  2 radians by 2 you get a definition for 1
radian:
180 

 1 radian
You can then work out a relationship for any number of radians (say n radians) by
multiplying both sides by n:
n  180 

 n radians
Want to know more?
If you have any further questions about this topic you can make an appointment to see a
Learning Enhancement Tutor in the Student Support Service, as well as speaking to
your lecturer or adviser.
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