Trigonometry is a crucial mathematical discipline that explores the relationships between angles and side lengths in triangles. Chapter 7 of Cambridge Mathematics delves into the significance of trigonometric functions and their applications in real-world scenarios, such as waves in physics and engineering. This chapter covers essential concepts including sine, cosine, and tangent functions, along with their graphs and complementary identities. It is designed for Year 11 students studying Mathematics Extension 1, providing practical problems and examples to enhance understanding. Key topics include the geometric foundations of trigonometry and its relevance in modern science and technology.

Key Points

  • Explains the fundamental concepts of trigonometric functions including sine, cosine, and tangent.
  • Covers practical applications of trigonometry in real-world scenarios such as waves and navigation.
  • Includes detailed examples and problems relevant for Year 11 Mathematics Extension 1 students.
  • Develops the relationship between angles and side lengths in triangles and circles.
Leo
Author:Pender et al.
Edition:2025
67 pages
Language:English
Type:Textbook
Trigonometry
Leo
Author:Pender et al.
Edition:2025
67 pages
Language:English
Type:Textbook
Trigonometry
149
/ 67
7
Trigonometry
Chapter introduction
Trigonometry is important in modern science principally because the graphs of the sine and cosine functions are
waves. Waves appear everywhere in the natural world, for example as water waves, as sound waves, or as the
electromagnetic waves that are responsible for radio, heat, light, ultraviolet radiation, X-rays and gamma rays.
In quantum mechanics, a wave is associated with every particle.
Trigonometry began in classical times, however, in practical situations such as building, surveying, navigation
and astronomy. Trigonometry uses the relationships between the angles and the side lengths in a triangle,
and its name comes from the Greek words trigonon, ‘triangle’, and metron, ‘measure’. This chapter develops
the trigonometric functions and their graphs from the geometry of triangles and circles, and applies the
trigonometric functions in practical problems.
Some of this chapter will be new to most readers, in particular the extension of the trigonometric functions to
angles of any magnitude, the graphs of these functions, and trigonometric identities and equations.
The graphs of y = cosec x, y = sec x and y = cot x are included here, in Section
7C, rather than in Chapter 6, so
that they take their place amongst the other three trig graphs. Some understanding of graphs of reciprocals —
presented in Chapter6—ishelpful, but readers who have not yet studied Chapter 6 should have little difficulty
following the presentation here..
Radian measure is essential later for the calculus of the trigonometric functions. This chapter, however, is
already long, and we have placed radian measure a little later in Chapter 11, after differentiation, when its
relevance can be explained.
CambridgeMATHS NSW Stage 6 – Mathematics Extension 1 ISBN 978-1-009-65480-7
Year 11 Photocop
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is restricted under law and this material must not be transferred to another part
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© Pender et al. 2025 Cambrid
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7A Trigonometry with right-angled triangles 241
7A
Trigonometry with right-angled triangles
Learning intentions
• Understand the role of Pythagoras’ theorem and similarity in trigonometry.
• Know the definitions of the six trigonometric functions for acute angles.
• Evaluate the trigonometric functions of the special angles 30
◦
,45
◦
and 60
◦
.
• Find sides and angles of a right-angled triangle, given sufficient information.
This section and the next will review the definitions of the trigonometric functions for acute angles, and apply
them to problems involving right-angled triangles.
Pythagoras’ theorem
You will know from earlier years that the trigonometry of triangles begins with two fundamental ideas —
Pythagoras’ theorem and similarity–congruence.
Pythagoras’ theorem tells us how to find the third side of a right-angled triangle. The theorem is the best-known
theorem in all of mathematics, and has been mentioned several times already in earlier chapters. Here is what it
says:
The square on the hypotenuse of a right-angled triangle is the sum of the squares on the other two
sides.
The diagrams below provide a very simple proof. Can you work out how the four shaded triangles have been
pushed around inside the square to prove the theorem?
B
A
C
c
c
B
A
C
a
a
b
b
Similarity and congruence
Similarity is required to define the trigonometric functions,
because each function is defined as the ratio of two sides of a
triangle.
• Two figures are called congruent if one can be obtained from
the other by translations, rotations and reflections.
• They are called similar if enlargements are allowed as well.
BC
A
6
12
10
P
QR5
6
3
In two similar figures:
matching angles are equal and matching sides are in ratio.
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
242 Chapter 7 Trigonometry
7A
The trigonometric functions for acute angles
Let θ be any acute angle, that is, 0
◦
< θ < 90
◦
. Construct a right-angled triangle with
an acute angle θ, and label the sides:
hyp — the hypotenuse, the side opposite the right angle,
opp — the side opposite the angle θ,
adj — the third side, adjacent to θ but not the hypotenuse.
θ
adj
opp
hyp
1 The trigonometric functions for an acute angle θ
sin θ =
opp
hyp
cos θ =
adj
hyp
tan θ =
opp
adj
cosec θ =
hyp
opp
sec θ =
hyp
adj
cot θ =
adj
opp
Any two triangles with angles of 90
◦
and θ are similar, by the AA similarity test. Hence the values of the six
trigonometric functions are the same, whatever the size of the triangle. The full names of the six trigonometric
functions are:
sine, cosine, tangent, cosecant, secant, cotangent.
Question
19 in the next exercise gives some clues about these names, and the way in which the functions were
originally defined.
Special angles
The values of the trigonometric functions for the three acute angles 30
◦
,45
◦
and 60
◦
can be calculated exactly,
using half a square and half an equilateral triangle, and applying Pythagoras’ theorem.
Ö2
45º
45º
1
1
CB
A
30º
60º
2
1
3
P
QR
Take half a square with side length 1 .
The resulting right-angled triangle ABC has two angles
of 45
◦
.
By Pythagoras’ theorem, the hypotenuse AC has
length
√
2.
Take half an equilateral triangle with side length 2 by
dropping an altitude.
The resulting right-angled
PQR has angles of 60
◦
and 30
◦
.
By Pythagoras’ theorem, PQ =
√
3.
Applying the definitions on the previous page gives the values in the following table.
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
.
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End of Document
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FAQs

What are the primary trigonometric functions covered in this chapter?
This chapter focuses on the primary trigonometric functions: sine, cosine, and tangent. Each function is defined in relation to the angles and sides of a right triangle. The chapter also explores their graphs, which illustrate how these functions behave over different angles. Additionally, complementary identities such as sine and cosine relationships are discussed, providing a deeper understanding of their interconnections.
How does this chapter relate trigonometry to real-world applications?
Trigonometry is applied in various fields, including physics, engineering, and astronomy. This chapter highlights how trigonometric functions model waves, which are fundamental in understanding sound, light, and other electromagnetic phenomena. By connecting theoretical concepts to practical applications, students can appreciate the relevance of trigonometry in modern science.
What types of problems can students expect to solve in this chapter?
Students will encounter a variety of problems that require applying trigonometric functions to find unknown side lengths and angles in triangles. The chapter includes practical scenarios, such as calculating distances across rivers and using the cosine rule in isosceles triangles. These problems are designed to reinforce the concepts learned and develop problem-solving skills.
What is the significance of the graphs of trigonometric functions?
The graphs of trigonometric functions, such as sine and cosine, are essential for visualizing how these functions behave over a range of angles. Understanding these graphs helps students grasp concepts like periodicity and amplitude, which are crucial in applications involving waves. The chapter emphasizes the importance of these graphs in interpreting real-world phenomena.
How does this chapter prepare students for advanced mathematics?
This chapter lays a strong foundation in trigonometry, which is essential for advanced studies in mathematics and science. By mastering the concepts and applications of trigonometric functions, students will be better equipped for higher-level topics such as calculus and physics. The skills developed in this chapter are critical for success in future mathematical endeavors.
What geometric concepts are foundational to understanding trigonometry?
Understanding the properties of triangles and circles is fundamental to trigonometry. This chapter begins with the geometric definitions of trigonometric functions based on right triangles, then extends these concepts to include the unit circle. This geometric perspective is crucial for grasping how trigonometric functions relate to angles and side lengths.
What are complementary identities in trigonometry?
Complementary identities in trigonometry refer to relationships between trigonometric functions of complementary angles, such as sine and cosine. For example, the identity sin(90° - θ) = cos(θ) illustrates how the sine of an angle is equal to the cosine of its complement. This chapter explains these identities and their significance in simplifying trigonometric expressions.

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