Lesson 1 ~35 min Unit 3 · Trigonometry +85 XP

Introduction to Trigonometric Ratios

Discover the three ratios that unlock every right-angled triangle. Sine, cosine and tangent connect angles to sides in ways that power navigation, engineering and astronomy.

Today's hook: Every time a builder checks a roof pitch or a pilot calculates a descent, they're using the same three ratios you'll learn today. Why do angles and sides stay locked together?

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Think First

warm-up

A ladder leans against a wall. If you know the angle the ladder makes with the ground and the length of the ladder, what else could you work out? How?

Record your answer in your workbook.

The Big Idea

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For any right-angled triangle, the ratios of sides depend only on the angles -- not the size. A small triangle and a large triangle with the same angles share identical sine, cosine and tangent values. That's why one ratio table works for every triangle in the world.

Draw two right-angled triangles with the same angles but different sizes. Measure opposite ÷ hypotenuse for both. The answer is identical. This remarkable fact is the foundation of trigonometry.

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$

Same angle, same ratio

If two triangles share the same angles, their trig ratios are identical.

Size doesn't matter

Enlarging a triangle changes the sides, but not the ratios between them.

One table, every triangle

That's why a calculator can store every trig value in one lookup table.

What You'll Master

objectives

Know

The names of the three sides relative to a marked angle
The definitions of sine, cosine and tangent
The mnemonic SOH CAH TOA

Understand

How the ratios are derived from similar right-angled triangles
That each ratio links one angle to two specific sides
Why exact values for 30°, 45° and 60° are worth memorising

Can Do

Label the opposite, adjacent and hypotenuse for any marked angle
Write the three trig ratios for a given right-angled triangle
Recall exact values without a calculator

Words You Need

vocabulary

HypotenuseThe longest side of a right-angled triangle, opposite the right angle.

OppositeThe side directly across from the marked angle θ.

AdjacentThe side next to the marked angle θ that is not the hypotenuse.

SOH CAH TOAMnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

RatioA comparison of two quantities by division. Here, one side divided by another.

Exact valueA value expressed as a fraction or surd rather than a rounded decimal.

Spot the Trap

heads-up

Wrong: "The adjacent is always the bottom side."

Right: The adjacent depends on which angle is marked. Rotate the triangle and the labels change.

Wrong: "$\sin \theta = \frac{\text{hypotenuse}}{\text{opposite}}$"

Right: SOH = $\frac{\text{Opposite}}{\text{Hypotenuse}}$ -- the side opposite the angle goes on top.

Parts of the Whole

+5 XP

Every right-angled triangle has three sides. The Hypotenuse is fixed -- it is always opposite the right angle and always the longest. The other two names depend entirely on which angle is marked.

The hypotenuse never changes. The opposite and adjacent are defined relative to the marked angle θ. Rotate the triangle or mark a different angle and those two labels swap.

$\text{H}^2 = \text{O}^2 + \text{A}^2$

Hypotenuse never changes

Always opposite the right angle. Always the longest side.

Opposite is across

Look directly across from the marked angle.

Adjacent is next door

Next to the angle, but not the hypotenuse.

Try It Now: In a right triangle with angle θ at the bottom left and the right angle at the bottom right, which side is the adjacent? (Hint: it's next to θ and not the hypotenuse.) Answer: the bottom side.

SOH CAH TOA

+5 XP

The three ratios connect one angle to two sides. Each ratio is a fraction with a specific pair of sides. The mnemonic SOH CAH TOA locks them into memory.

Sine uses Opposite and Hypotenuse. Cosine uses Adjacent and Hypotenuse. Tangent uses Opposite and Adjacent. Say the mnemonic aloud until it sticks.

$\sin \theta = \frac{O}{H}$
$\cos \theta = \frac{A}{H}$
$\tan \theta = \frac{O}{A}$

Opposite over Hypotenuse

Sine. The side across from the angle goes on top.

Adjacent over Hypotenuse

Cosine. The side next to the angle over the longest side.

Opposite over Adjacent

Tangent. The steepness ratio -- opposite divided by adjacent.

Exact Values

+5 XP

Two special triangles produce exact values for 30°, 45° and 60°. These bypass the calculator and show up constantly in tests. Memorise them once, use them forever.

The 45-45-90 triangle has two equal legs of 1 and hypotenuse √2. The 30-60-90 triangle has sides 1, √3 and 2. From these two shapes every exact trig value is derived.

$\sin 30° = \frac{1}{2}$

45° has equal legs

Two sides of 1, hypotenuse √2. Sine and cosine are identical.

30° and 60° swap

sin 30° = cos 60° and cos 30° = sin 60°.

Memorise the table

Tests love exact values. No calculator needed.

Exact values table -- worth copying into your books:

Angle	$\sin$	$\cos$	$\tan$
$30°$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$
$45°$	$\frac{1}{\sqrt{2}}$	$\frac{1}{\sqrt{2}}$	$1$
$60°$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$

Write the Ratios

+5 XP

Once the sides are labelled, writing the ratios is just reading. Match the sides to the mnemonic and write each as a fraction in simplest form.

Start by identifying Opposite, Adjacent and Hypotenuse. Then match each ratio to its pair of sides. Simplify the fraction if possible.

$\sin \theta = \frac{3}{5}$

Label before you ratio

Never write a ratio until you have named all three sides.

Fractions in lowest terms

$\frac{6}{10}$ should be simplified to $\frac{3}{5}$.

Sine is always less than 1

The hypotenuse is the longest side, so O/H and A/H are always < 1.

Watch Me Solve It · 3 examples

Watch Me Solve It · Labelling sides

+15 XP per step

PROBLEM

A right-angled triangle has angle θ marked at the bottom left and the right angle at the top left. Label the hypotenuse, opposite and adjacent relative to θ.

1

Identify the right angle

Right angle is at the top left corner.

The little square symbol marks the 90° angle.
2

Find the hypotenuse

Hypotenuse = the diagonal side on the right.

The hypotenuse is always opposite the right angle.
3

Find the opposite

Opposite = the vertical side on the right.

Look directly across from the marked angle θ.
4

Find the adjacent

Adjacent = the horizontal bottom side.

Next to θ, but not the hypotenuse.

AnswerHypotenuse = diagonal, Opposite = vertical right, Adjacent = horizontal bottom

Watch Me Solve It · Writing the ratios

+15 XP per step

PROBLEM

For a right-angled triangle with angle θ, opposite = 8 cm, adjacent = 15 cm and hypotenuse = 17 cm. Write $\sin \theta$, $\cos \theta$ and $\tan \theta$.

1

State the three sides

O = 8, A = 15, H = 17

Label first so you don't mix up the numbers.
2

Apply SOH

$\sin \theta = \frac{O}{H} = \frac{8}{17}$
3

Apply CAH

$\cos \theta = \frac{A}{H} = \frac{15}{17}$
4

Apply TOA

$\tan \theta = \frac{O}{A} = \frac{8}{15}$

All fractions are already in lowest terms.

Answer$\sin \theta = \frac{8}{17}$, $\cos \theta = \frac{15}{17}$, $\tan \theta = \frac{8}{15}$

Watch Me Solve It · Exact values

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PROBLEM

Write the exact value of (a) $\sin 30°$, (b) $\cos 45°$, (c) $\tan 60°$. No calculator.

1

Use the 30-60-90 triangle for sin 30°

$\sin 30° = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{2}$

The side opposite 30° is 1; the hypotenuse is 2.
2

Use the 45-45-90 triangle for cos 45°

$\cos 45° = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{2}}$

Both legs are 1; the hypotenuse is $\sqrt{2}$.
3

Use the 30-60-90 triangle for tan 60°

$\tan 60° = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{3}}{1} = \sqrt{3}$

Opposite 60° is $\sqrt{3}$; adjacent is 1.

Answer(a) $\frac{1}{2}$ · (b) $\frac{1}{\sqrt{2}}$ · (c) $\sqrt{3}$

Common Pitfalls

heads-up

Thinking adjacent is always the bottom side

Adjacent means "next to the marked angle" -- not "bottom". Rotate the triangle and the adjacent could be vertical.

Fix: always ask "Which angle is marked?" before labelling opposite and adjacent.

Swapping sine and cosine

$\sin \theta = \frac{A}{H}$ and $\cos \theta = \frac{O}{H}$ is a common mix-up. The mnemonic SOH CAH TOA exists to prevent this.

Fix: Opposite goes with Sine (both start with a vowel-ish O sound). Adjacent goes with Cosine.

Forgetting that opposite and adjacent depend on the marked angle

If you mark the other acute angle, opposite and adjacent swap places. The hypotenuse stays put.

Fix: re-label O and A every time the marked angle changes.

Copy Into Your Books

SOH CAH TOA

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}}$
$\cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}}$
$\tan \theta = \frac{\text{Opposite}}{\text{Adjacent}}$

Side Labels

Hypotenuse -- opposite right angle, always longest
Opposite -- across from marked angle
Adjacent -- next to marked angle, not hypotenuse

Exact Values

sin 30° = 1/2, cos 30° = √3/2, tan 30° = 1/√3
sin 45° = 1/√2, cos 45° = 1/√2, tan 45° = 1
sin 60° = √3/2, cos 60° = 1/2, tan 60° = √3

Similar Triangles

Same angles = same trig ratios
Size does not affect the ratios
One table works for every triangle

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Mixed

4 problems

Four problems mixing side labelling, ratio writing and exact values. Work each one, then reveal the answer.

1 In a right triangle with angle θ, which side is the hypotenuse?

The side opposite the right angle. It is always the longest side.
2 Write $\sin \theta$ if opposite = 5 and hypotenuse = 13.

$\sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{5}{13}$$= \frac{5}{13}$
3 A right triangle has sides 6, 8, 10. Angle θ is opposite side 6. Write all three ratios.

O = 6, A = 8, H = 10. Simplify: $\sin \theta = \frac{3}{5}$, $\cos \theta = \frac{4}{5}$, $\tan \theta = \frac{3}{4}$
4 What is the exact value of $\cos 60°$?

In the 30-60-90 triangle, adjacent to 60° = 1 and hypotenuse = 2.$\cos 60° = \frac{1}{2}$

Complete in your workbook.

Quick Check · 5 questions

Which side is opposite θ?

+10 XP

Which ratio is $\sin \theta$?

+10 XP

$\tan \theta = \frac{3}{4}$ and opposite = 6. Find adjacent.

+10 XP

What is $\cos 60°$?

+10 XP

If $\sin \theta = \frac{5}{13}$ and $\cos \theta = \frac{12}{13}$, find $\tan \theta$.

+10 XP

Show Your Working · 3 questions

Show Your Working

7 marks total

Apply Easy 2 MARKS

Q6. For a right-angled triangle with angle θ, opposite = 7 cm, adjacent = 24 cm and hypotenuse = 25 cm. Write $\sin \theta$, $\cos \theta$ and $\tan \theta$.

Answer in your workbook.

Apply Medium 2 MARKS

Q7. A right-angled triangle has sides in the ratio $3:4:5$. Let θ be the angle opposite the side of length 3. (a) Write exact values for $\sin \theta$, $\cos \theta$ and $\tan \theta$. (b) Explain why $\tan \theta$ must be less than 1.

Answer in your workbook.

Analyse Hard 3 MARKS

Q8. Two similar right triangles A and B both have angles 30°, 60°, 90°. Triangle A has hypotenuse 10 cm. Triangle B has hypotenuse 25 cm. (a) Find the scale factor from A to B. (b) Find the side opposite 30° in B. (c) Explain why $\sin 30°$ is the same for both triangles.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C -- The opposite is directly across from the marked angle.

2. B -- SOH: Sine = Opposite / Hypotenuse.

3. C -- $\frac{3}{4} = \frac{6}{x}$ gives $x = 8$.

4. B -- cos 60° = 1/2 from the 30-60-90 triangle.

5. B -- $\tan \theta = \frac{O}{A} = \frac{5}{12}$.

Show Your Working Model Answers

Q6 (2 marks): $\sin \theta = \frac{7}{25}$, $\cos \theta = \frac{24}{25}$, $\tan \theta = \frac{7}{24}$ [2].

Q7 (2 marks): (a) $\sin \theta = \frac{3}{5}$, $\cos \theta = \frac{4}{5}$, $\tan \theta = \frac{3}{4}$ [1]. (b) $\tan \theta = \frac{3}{4} < 1$ because the opposite side (3) is shorter than the adjacent side (4) [1].

Q8 (3 marks): (a) Scale factor = $\frac{25}{10} = 2.5$ [1]. (b) Opposite 30° in A = 5 cm, so in B = $5 \times 2.5 = 12.5$ cm [1]. (c) Trig ratios depend only on the angles, not the size of the triangle. Similar triangles have identical angles, so the ratios are the same [1].

Stretch Challenge · +25 XP, +10 coins

The Pythagorean Identity

Prove that $\sin^2 \theta + \cos^2 \theta = 1$ for any right-angled triangle with opposite side $a$, adjacent side $b$ and hypotenuse $c$. Start by writing $\sin \theta$ and $\cos \theta$ as fractions, then add them.

Reveal solution

$\sin \theta = \frac{a}{c}$ and $\cos \theta = \frac{b}{c}$. So $\sin^2 \theta + \cos^2 \theta = \frac{a^2}{c^2} + \frac{b^2}{c^2} = \frac{a^2 + b^2}{c^2}$. By Pythagoras, $a^2 + b^2 = c^2$, so $\frac{c^2}{c^2} = 1$.

Quick Review

Label first

Name O, A and H before writing any ratio

SOH

$\sin \theta = \frac{O}{H}$

CAH

$\cos \theta = \frac{A}{H}$

TOA

$\tan \theta = \frac{O}{A}$

Exact values

Memorise 30°, 45°, 60°

Size doesn't matter

Same angles = same ratios

Interactive: SOH CAH TOA Labeller

Practise labelling the sides and writing the ratios on random triangles until it's automatic.

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DAILY CHALLENGE · TODAY

The Ratio Sprint

Label the sides and write all three ratios for a triangle with opposite = 5, adjacent = 12 and hypotenuse = 13 in under 60 seconds for double coins!

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Lesson 2 -- Finding Unknown Sides

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