← Unit 4 Introduction to Trigonometry

Printable Worksheets

Print or save as PDF, or build a custom worksheet from any module's questions.

Build Apply Master Build custom
MA5-TRG-C-01

Introduction to Trigonometry

⏱ 25 min📚 Year 9📈
Think First

If a right-angled triangle has angles 30° and 60°, what do you know about its sides?

💡 Revision: Ensure you understand angles, triangles, and Pythagoras theorem.

Learning Intentions

Know

  • Right-angled triangles
  • Opposite, adjacent, hypotenuse
  • Trigonometric ratios

Understand

  • Why trig ratios depend only on the angle
  • How similar triangles produce consistent ratios

Can Do

  • Label sides of a right-angled triangle
  • Identify the hypotenuse
  • Set up trig ratios from a diagram
HypotenuseOppositeAdjacentRight angle
Learn Phase
1

Right-Angled Triangles

The foundation of trigonometry

A right-angled triangle has one 90° angle. The side opposite the right angle is the hypotenuseit is always the longest side.

Given an angle $\theta$ (other than the right angle):

  • Opposite: the side across from $\theta$
  • Adjacent: the side next to $\theta$ (not the hypotenuse)

Example: In a triangle with angle $30°$, the side opposite $30°$ is half the hypotenuse.

2

The Trigonometric Ratios

SOH CAH TOA

For any angle $\theta$ in a right-angled triangle:

Trigonometric Ratios

$sin \theta = dfrac{\text{opposite}}{\text{hypotenuse}}$

$cos \theta = dfrac{\text{adjacent}}{\text{hypotenuse}}$

$\tan \theta = dfrac{\text{opposite}}{\text{adjacent}}$

Mnemonic: SOH CAH TOA

  • Sin = Opposite / Hypotenuse
  • Cos = Adjacent / Hypotenuse
  • Tan = Opposite / Adjacent
3

Using a Calculator

Finding values of trig ratios

Use your calculator to find trig values:

  • Make sure it is in degree mode
  • $sin 30° = 0.5$
  • $cos 45° approx 0.707$
  • $\tan 60° approx 1.732$

To find an angle from a ratio, use inverse functions: $sin^{-1}$, $cos^{-1}$, $\tan^{-1}$.

Example: If $sin \theta = 0.5$, then $\theta = sin^{-1}(0.5) = 30°$.

✓

Check Understanding

Try it yourself

In a right-angled triangle with angle $40°$, label the opposite, adjacent, and hypotenuse.

Worked Example

Introduction to Trigonometry

1

Find $sin 45°$, $cos 45°$, and $\tan 45°$.

$sin 45° = dfrac{1}{sqrt{2}} approx 0.707$

$cos 45° = dfrac{1}{sqrt{2}} approx 0.707$

$\tan 45° = 1$

2

A right-angled triangle has angle $30°$ and hypotenuse 10 cm. Find the opposite side.

$sin 30° = dfrac{\text{opposite}}{10}$

$\text{opposite} = 10 \times sin 30° = 10 \times 0.5 = 5$ cm

3

Find $\theta$ if $\tan \theta = 1$.

$\theta = \tan^{-1}(1) = 45°$

Common Misconceptions

✕

The hypotenuse is opposite the smallest angle. No, the hypotenuse is opposite the right angle (90°), which is the largest angle.

✕

$sin \theta$ can be greater than 1. No, since the opposite side is always shorter than the hypotenuse, $sin \theta$ is always between 0 and 1.

✕

$\tan 90° = 1$. No, $\tan 90°$ is undefined because the adjacent side would be zero.

Your Turn

Practice, Trigonometry Basics

Work through each question in your book or digitally. Answers are in the Questions phase.

1Label the sides of a right-angled triangle with angle $25°$.
2Find $sin 60°$ and $cos 30°$. What do you notice?
3Find $\theta$ if $sin \theta = 0.766$.
Real-World Anchor

Architecture and Engineering

The Sydney Opera House and many Australian bridges use trigonometry in their design. Engineers calculate forces, angles, and lengths using trigonometric ratios to ensure structures are safe and stable.

📓 Copy Into Your Books

▼

SOH CAH TOA

  • $sin = \text{opp}/\text{hyp}$
  • $cos = \text{adj}/\text{hyp}$
  • $\tan = \text{opp}/\text{adj}$

Calculator

  • Ensure degree mode
  • $sin^{-1}$, $cos^{-1}$, $\tan^{-1}$ for angles

Key Facts

  • Hypotenuse is longest side
  • Opposite is across from angle
  • Adjacent is next to angle
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
The hypotenuse is opposite the:
$sin 30° = $
In SOH CAH TOA, CAH means:
$\tan 45° = $
If $sin \theta = 0.5$, then $\theta = $
The side next to angle $\theta$ (not hypotenuse) is:
$cos 60° = $
$\tan 90°$ is:
1Find $sin 30°$, $cos 30°$, and $\tan 30°$ using exact values.
2A right-angled triangle has angle $40°$ and hypotenuse 12 cm. Find the adjacent side.
3Explain why $sin \theta$ cannot exceed 1.

Comprehensive Answers

1Label sides of triangle with angle $25°$.
Hypotenuse opposite right angle; opposite across from $25°$; adjacent next to $25°$.
2Find $sin 60°$ and $cos 30°$.
Both equal $sqrt{3}/2 approx 0.866$.
3Find $\theta$ if $sin \theta = 0.766$.
$\theta = sin^{-1}(0.766) approx 50°$.
MC 1Hypotenuse is opposite.
Right angle. Answer: B
MC 2$sin 30°$.
$0.5$. Answer: B
MC 3CAH meaning.
Cos = Adj/Hyp. Answer: B
MC 4$\tan 45°$.
$1$. Answer: C
MC 5$sin \theta = 0.5$.
$30°$. Answer: B
MC 6Side next to angle.
Adjacent. Answer: B
MC 7$cos 60°$.
$0.5$. Answer: A
MC 8$\tan 90°$.
Undefined. Answer: C
SA 1Exact values for $30°$.
$sin 30° = 1/2$, $cos 30° = sqrt{3}/2$, $\tan 30° = 1/sqrt{3}$.
SA 2Adjacent side with $40°$ and hyp 12 cm.
$\text{adj} = 12 \times cos 40° approx 9.19$ cm.
SA 3Why $sin \theta$ cannot exceed 1.
Opposite side is always shorter than or equal to hypotenuse, so ratio is at most 1.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.
Want help with Introduction to Trigonometry, Year 9 Unit 4?

Work through this topic 1-on-1 with an experienced HSC tutor.

Book a free session →