Radian Measure of Angles explores the concept of measuring angles using radians instead of degrees. It covers the fundamental principles of trigonometry, including the definitions of sine, cosine, and tangent functions in radian measure. The chapter also includes practical applications of radians in various fields such as physics and engineering. Students will learn how to convert between degrees and radians, evaluate trigonometric functions, and solve equations involving angles in radians. This resource is essential for high school students studying mathematics and preparing for exams.

Key Points

  • Defines angle size in radians and explains the conversion between degrees and radians.
  • Covers trigonometric functions of angles given in radians with examples.
  • Includes exercises for converting angles between degrees and radians.
  • Explains the applications of radians in real-world scenarios, such as physics.
Leo
26 pages
Language:English
Type:Textbook
Biology
Leo
26 pages
Language:English
Type:Textbook
Biology
146
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13
Radian measure of angles
Chapter introduction
Chapter 7 reviewed trigonometry, and developed it for all angles instead of only for acute angles, until the
wave-like graphs of the sine and cosine functions could be displayed. These sine and cosine waves are
fundamental in a vast variety of applications. They model such phenomena as sound waves, light and radio
waves, vibrating strings, and tides. Many populations oscillate with the seasons, the economy tends to move in
cycles, and in quantum mechanics, everything in the universe is some form of wave.
We will apply calculus to trigonometric functions next year, but before we can do that, we need to develop
a new way to measure angles. We shall use radians, which are based on circles and π — one revolution is
2π radians — rather than the very ancient Babylonian system of degrees, where one revolution is the number of
days in a year. This reliance on circles is hardly surprising, because Chapter 7 presented the general definition
of the trigonometric functions based on circles.
With radian measure using π for the trig functions, and with Euler’s number e as the appropriate base for
exponential and log functions, we will be ready next year to extend calculus to both groups of functions. Most
applications of calculus use functions that combine powers and these two groups of functions.
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518 Chapter 13 Radian measure of angles
13A
13A
Radian measure of angle size
Learning intentions
• Define angle size in radians, and convert between degrees and radians.
• Find trigonometric functions of angles given in radians, exactly and approximately.
The use of degrees to measure angles is based on astronomy. There are 360 days in the year — plus festival days
—so1
◦
is the angle through which the Sun moves against the fixed stars each day (with apologies to Copernicus
and Galileo).
Mathematics is far too general a discipline to be tied to events in our solar system. We now develop a new system
for measuring angles based on mathematics alone.
Radian measure of angle size
The size of an angle in radians is defined as the ratio of two lengths in a circle.
Given an angle with vertex O, construct a circle with centre O meeting the two arms
of the angle at A and B.
The size of ∠AOB in radians is the ratio of the arc length AB and the radius OA.
O
A
B
r
r
1 Radian measure
∠AOB =
arc length AB
radius OA
• This definition gives the same angle size, whatever the radius of the circle, because all circles are
similar
to one another.
• Angle size is the ratio of two lengths, so is a pure number with no dimension.
Hence
the unit ‘radians’ is not required when giving an angle size in radians.
This definition of angle size is very similar to the definitions of the six trigonometric functions, which are also
ratios of lengths and so are also pure numbers.
Some basic conversion between degrees and radians
• The arc subtended by a revolution is the whole circumference,
so 1 revolution =
circumference
radius
=
2πr
r
= 2π.
A
O
r
• A straight angle subtends a semicircle,
so 1 straight angle =
arc length of semicircle
radius
=
πr
r
= π.
A
O
rr
B
CambridgeMATHS NSW Stage 6 – Mathematics Extension 1 ISBN 978-1-009-65480-7
Year 11 Photocop
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in
g
is restricted under law and this material must not be transferred to another part
y
.
© Pender et al. 2025 Cambrid
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13A Radian measure of angle size 519
• A right angle subtends a quarter-circle,
so 1 right angle =
arc length of quarter-circle
radius
=
1
2
πr
r
=
π
2
.
A
O
r
r
B
These three basic conversions should be memorised very securely.
2 Basic conversions between degrees and radian measure
360
◦
= 2π 180
◦
= π 90
◦
=
π
2
Because 180
◦
= π, an angle size in radians can be converted to an angle size in degrees by multiplying by
180
π
◦
.
Conversely, degrees are converted to radians by multiplying by
π
180
.
3 Converting between degrees and radians
• To convert from radians to degrees,
multiply by
180
π
◦
• To convert from degrees to radians,
multiply by
π
180
• One radian and one degree:
1
radian =
180
π
◦
57
◦
18
and 1 degree =
π
180
0.0175.
One radian is by definition the angle subtended at the centre of a circle by an arc of
length equal to the radius.
Notice that the sector OAB in the diagram to the right is almost an equilateral
triangle, so 1 radian is about 60
◦
. This makes sense of the value given above, that
1 radian is about 57
◦
.
A
B
r
r
r
O
1radian
Example 1 Converting degrees to radians
Express these angle sizes in radians.
60
◦
a 270
◦
b 495
◦
c 37
◦
d
Solution
60
◦
= 60 ×
π
180
=
π
3
a 270
◦
= 270 ×
π
180
=
3π
2
b
495
◦
= 495 ×
π
180
=
11π
4
c 37
◦
= 37 ×
π
180
=
37π
180
d
CambridgeMATHS NSW Stage 6 – Mathematics Extension 1 ISBN 978-1-009-65480-7
Year 11 Photocop
y
in
g
is restricted under law and this material must not be transferred to another part
y
.
© Pender et al. 2025 Cambrid
g
e Universit
y
Press & Assessment
/ 26
End of Document
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FAQs

What is the significance of radians in trigonometry?
Radians provide a natural way to measure angles based on the radius of a circle. One radian is defined as the angle subtended at the center of a circle by an arc whose length is equal to the radius. This relationship simplifies many mathematical calculations, especially in calculus and physics, where circular motion and periodic functions are involved. Understanding radians is crucial for accurately interpreting and solving problems related to trigonometric functions.
How do you convert degrees to radians?
To convert degrees to radians, multiply the degree measure by π/180. For example, to convert 60 degrees to radians, you would calculate 60 × π/180, which simplifies to π/3 radians. This conversion is essential for working with trigonometric functions that require angle measures in radians, especially in advanced mathematics and physics applications.
What are the main trigonometric functions defined in radians?
The main trigonometric functions defined in radians include sine (sin), cosine (cos), and tangent (tan). These functions are periodic and have specific values at key angles such as π/6, π/4, and π/3. Understanding how to evaluate these functions in radians is vital for solving trigonometric equations and applying these concepts in real-world scenarios, such as wave motion and oscillations.
What exercises are included in the chapter on radians?
The chapter includes various exercises that focus on converting angles between degrees and radians, evaluating trigonometric functions for angles in radians, and solving trigonometric equations. These exercises are designed to reinforce the concepts learned and provide practical applications of radians in mathematical problems. Additionally, students can practice using calculators set to radians mode to ensure accurate results.

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