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Introduction to Trigonometry
If a right-angled triangle has angles 30° and 60°, what do you know about its sides?
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
Right-Angled Triangles
A right-angled triangle has one 90° angle. The side opposite the right angle is the hypotenuse — it is always the longest side.
Given an angle $ heta$ (other than the right angle):
- Opposite: the side across from $ heta$
- Adjacent: the side next to $ heta$ (not the hypotenuse)
Example: In a triangle with angle $30°$, the side opposite $30°$ is half the hypotenuse.
The Trigonometric Ratios
For any angle $ heta$ in a right-angled triangle:
$sin heta = dfrac{ ext{opposite}}{ ext{hypotenuse}}$
$cos heta = dfrac{ ext{adjacent}}{ ext{hypotenuse}}$
$ an heta = dfrac{ ext{opposite}}{ ext{adjacent}}$
Mnemonic: SOH CAH TOA
- Sin = Opposite / Hypotenuse
- Cos = Adjacent / Hypotenuse
- Tan = Opposite / Adjacent
Using a Calculator
Use your calculator to find trig values:
- Make sure it is in degree mode
- $sin 30° = 0.5$
- $cos 45° approx 0.707$
- $ an 60° approx 1.732$
To find an angle from a ratio, use inverse functions: $sin^{-1}$, $cos^{-1}$, $ an^{-1}$.
Example: If $sin heta = 0.5$, then $ heta = sin^{-1}(0.5) = 30°$.
Check Understanding
In a right-angled triangle with angle $40°$, label the opposite, adjacent, and hypotenuse.
Introduction to Trigonometry
Find $sin 45°$, $cos 45°$, and $ an 45°$.
$sin 45° = dfrac{1}{sqrt{2}} approx 0.707$
$cos 45° = dfrac{1}{sqrt{2}} approx 0.707$
$ an 45° = 1$
A right-angled triangle has angle $30°$ and hypotenuse 10 cm. Find the opposite side.
$sin 30° = dfrac{ ext{opposite}}{10}$
$ ext{opposite} = 10 imes sin 30° = 10 imes 0.5 = 5$ cm
Find $ heta$ if $ an heta = 1$.
$ heta = an^{-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.
$sin heta$ can be greater than 1. No — since the opposite side is always shorter than the hypotenuse, $sin heta$ is always between 0 and 1.
$ an 90° = 1$. No — $ an 90°$ is undefined because the adjacent side would be zero.
Practice — Trigonometry Basics
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
- $sin = ext{opp}/ ext{hyp}$
- $cos = ext{adj}/ ext{hyp}$
- $ an = ext{opp}/ ext{adj}$
Calculator
- Ensure degree mode
- $sin^{-1}$, $cos^{-1}$, $ an^{-1}$ for angles
Key Facts
- Hypotenuse is longest side
- Opposite is across from angle
- Adjacent is next to angle