Lesson 6 ~25 min Unit 3 · Trigonometry +85 XP

Introducing Trigonometric Ratios

Meet sin, cos and tan, the three ratios that connect a right-angled triangle's angle $\theta$ to its sides. Memorise SOH-CAH-TOA.

Today's hook: Any right-angled triangle with a 30° angle, whether tiny or huge, has EXACTLY the same ratio of opposite side to hypotenuse: 0.5. Why is that ratio always the same?

0/5QUESTS

Printable Worksheets

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

Build Apply Master Build custom

Think First

warm-up

Two right triangles both have an angle of 40°. One has hypotenuse 5 cm, the other 50 cm. Will the ratio ‘opposite ÷ hypotenuse’ be the same in both? Why or why not?

Record your answer in your workbook.

The Big Idea

+5 XP

For any right triangle, the THREE side-pair ratios (opp/hyp, adj/hyp, opp/adj) depend ONLY on the angle $\theta$, not on the size. We give them names: sine, cosine and tangent.

$\sin\theta = \dfrac{\text{opp}}{\text{hyp}}$, $\cos\theta = \dfrac{\text{adj}}{\text{hyp}}$, $\tan\theta = \dfrac{\text{opp}}{\text{adj}}$. Memorise these with SOH-CAH-TOA: Sine = Opp/Hyp, Cosine = Adj/Hyp, Tangent = Opp/Adj. Same angle → same ratio, every time.

SOH-CAH-TOA: Sin=Opp/Hyp, Cos=Adj/Hyp, Tan=Opp/Adj

Same angle, same ratio

Doubling the triangle doubles all sides, ratios stay constant.

Three ratios only

sin, cos, tan, that's it. They're the three pairwise ratios.

SOH-CAH-TOA

The memory phrase: $\sin = $O/H, $\cos = $A/H, $\tan = $O/A.

What You'll Master

objectives

Know

The three trig ratios: sin, cos, tan
SOH-CAH-TOA mnemonic
Ratios depend on the angle only, not triangle size

Understand

Why similar triangles (same angles) share trig ratios
How sin and cos always lie between 0 and 1 for acute angles
Why tan can exceed 1 (when opp > adj)

Can Do

Write each trig ratio in terms of opp, adj, hyp
Calculate sin, cos, tan given two sides
Recall SOH-CAH-TOA fluently

Words You Need

vocabulary

Sine ($\sin$)The ratio opposite/hypotenuse for a given angle. Always between 0 and 1 for acute angles.

Cosine ($\cos$)The ratio adjacent/hypotenuse. Always between 0 and 1 for acute angles.

Tangent ($\tan$)The ratio opposite/adjacent. Can be any positive value for acute angles, even much bigger than 1.

RatioA comparison of two quantities by division. Has no units when both quantities are lengths in the same unit.

SOH-CAH-TOAMemory aid: $\sin$=Opp/Hyp, $\cos$=Adj/Hyp, $\tan$=Opp/Adj.

Similar trianglesTriangles with the same angles but possibly different sizes. They share identical trig ratios.

Spot the Trap

heads-up

Wrong: “$\sin 30° = $ something different in a bigger triangle.” No, same angle, same sine, always.

Right: $\sin 30° = 0.5$ in EVERY right triangle that contains a 30° angle.

Wrong: “tan can't be bigger than 1.” Wrong, tan goes to infinity as the angle approaches 90°.

Right: For acute $\theta$: sin and cos are in $[0, 1]$, but tan can be any positive number.

Reading the Ratios

+5 XP

To find a trig ratio from a triangle, identify the three sides relative to $\theta$, then choose the right formula.

With sides labelled opp, adj, hyp (relative to $\theta$), compute the ratio: $\sin\theta$ uses opp and hyp, $\cos\theta$ uses adj and hyp, $\tan\theta$ uses opp and adj. Notice that hyp appears in sin and cos but NOT in tan.

sin needs hyp, cos needs hyp, tan does NOT

Identify $\theta$

Mark $\theta$ first, then label opp/adj/hyp.

sin + cos pair

Both use hyp on the bottom, only the top differs.

tan is special

Tan needs both legs, no hyp.

Why Ratios Don't Depend on Size

+5 XP

If you double a right triangle, you double every side, but the ratios opp/hyp, adj/hyp, opp/adj stay exactly the same. This is the magic of similar triangles.

Triangle	opp	hyp	opp/hyp
3-4-5	3	5	0.6
6-8-10	6	10	0.6
15-20-25	15	25	0.6

Each triangle has the same acute angle, so the same $\sin\theta = 0.6$.

Scale the triangle → ratios unchanged

Same angle = same ratio

Any two right triangles sharing the angle $\theta$ have the same sin, cos, tan of $\theta$.

Calculator stores them

Your calculator remembers every sin, cos, tan value for you.

Greek letter

$\theta$ is just a variable name, like $x$ for length.

Watch Me Solve It · 3 examples

Watch Me Solve It · Compute all three ratios

+15 XP per step

PROBLEM

In a right triangle, opp = 3, adj = 4, hyp = 5. Find $\sin\theta$, $\cos\theta$, $\tan\theta$.

1

Sine

$\sin\theta = \dfrac{\text{opp}}{\text{hyp}} = \dfrac{3}{5} = 0.6$
2

Cosine

$\cos\theta = \dfrac{\text{adj}}{\text{hyp}} = \dfrac{4}{5} = 0.8$
3

Tangent

$\tan\theta = \dfrac{\text{opp}}{\text{adj}} = \dfrac{3}{4} = 0.75$

Answer$\sin\theta=0.6$, $\cos\theta=0.8$, $\tan\theta=0.75$

Watch Me Solve It · Same angle, scaled triangle

+15 XP per step

PROBLEM

A second triangle with the same angle $\theta$ has opp = 9, adj = 12, hyp = 15. Show $\tan\theta$ is the same.

1

Compute

$\tan\theta = 9/12$
2

Simplify

$= 3/4 = 0.75$
3

Conclude

Same as 3-4-5 triangle.

This triangle is just 3$\times$(3-4-5), same angles, same trig ratios.

Answer$\tan\theta = 0.75$, same as the smaller triangle.

Watch Me Solve It · Use SOH-CAH-TOA to choose

+15 XP per step

PROBLEM

A right triangle has opp = 8 and adj = 6 (with respect to $\theta$). Which trig ratio relates these two sides only? Compute it.

1

Identify ratio

opp/adj is the tangent definition.
2

Apply

$\tan\theta = 8/6 = 4/3$
3

Approx

$\approx 1.33$

Answer$\tan\theta = 4/3 \approx 1.33$

Common Pitfalls

heads-up

Swapping opp and adj in tan

Writing $\tan\theta = $ adj/opp instead of opp/adj.

Fix: TOA, Tangent = Opposite/Adjacent. Opp is on TOP.

Using hyp in tan

Putting the hypotenuse into a tan ratio.

Fix: Tan uses only the two legs, never the hypotenuse.

Mixing up which side is which

Calling the wrong side opposite or adjacent.

Fix: Identify $\theta$, find the side ACROSS = opp, find the leg NEXT = adj.

Copy Into Your Books

SOH-CAH-TOA

$\sin\theta = $ Opp/Hyp
$\cos\theta = $ Adj/Hyp
$\tan\theta = $ Opp/Adj

Ratios

sin: opp & hyp
cos: adj & hyp
tan: opp & adj

Properties

Same angle → same ratio
sin, cos $\in [0,1]$ for acute $\theta$
tan can be very large

No hyp in tan

Tan uses both legs
Doesn't need hyp
Useful when no hyp known

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · SOH-CAH-TOA

4 problems

Four quick drills to lock in today's skill. Try each, then reveal the answer.

1 $\sin\theta = $?

Sine = Opp / Hyp.Opp / Hyp
2 $\cos\theta = $?

Cosine = Adj / Hyp.Adj / Hyp
3 $\tan\theta = $?

Tangent = Opp / Adj.Opp / Adj
4 opp = 5, hyp = 13. Find $\sin\theta$.

$\sin\theta = 5/13$.$\sin\theta = 5/13 \approx 0.385$

Complete in your workbook.

Quick Check · 5 questions

$\sin\theta$ equals:

+10 XP

In a 3-4-5 triangle with $\theta$ opposite the side of length 3, $\cos\theta = ?$

+10 XP

Which ratio does NOT use the hypotenuse?

+10 XP

opp = 7, adj = 24, hyp = 25. Find $\tan\theta$.

+10 XP

Two right triangles share an angle of 40°. Which is true about $\sin 40°$?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. In a right triangle, opp = 6, adj = 8, hyp = 10 with respect to $\theta$. Calculate (a) $\sin\theta$, (b) $\cos\theta$, (c) $\tan\theta$ as decimals.

Answer in your workbook.

UnderstandMedium2 MARKS

Q7. State SOH-CAH-TOA and explain in your own words what each part means.

Answer in your workbook.

ReasonHard4 MARKS

Q8. Triangle A has sides 5, 12, 13 (with $\theta$ at the angle opposite 5). Triangle B has sides 15, 36, 39 (with $\theta$ at the angle opposite 15). Show that $\sin\theta$ is identical in both triangles, and explain why.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C SOH = Sine = Opp/Hyp.

2. B adj=4, hyp=5.

3. D Tan = opp/adj, no hyp.

4. A$\tan\theta = 7/24$.

5. B Same angle → same sine.

Show Your Working Model Answers

Q6 (3 marks): (a) $\sin\theta = 6/10 = 0.6$ [1]. (b) $\cos\theta = 8/10 = 0.8$ [1]. (c) $\tan\theta = 6/8 = 0.75$ [1].

Q7 (2 marks): S-O-H: Sine equals Opposite over Hypotenuse [1/2]. C-A-H: Cosine equals Adjacent over Hypotenuse [1/2]. T-O-A: Tangent equals Opposite over Adjacent [1]. Together they give the three trig ratios for any angle in a right triangle.

Q8 (4 marks): $\sin\theta_A = 5/13 \approx 0.385$ [1]. $\sin\theta_B = 15/39 = 5/13 \approx 0.385$ [1]. Identical [1]. Triangle B is $3\times$ triangle A, every side is scaled by 3, so opp/hyp = $3\cdot 5/(3\cdot 13) = 5/13$. The scaling cancels in the ratio, leaving the same value. Same angles → same trig ratios [1].

Stretch Challenge · +25 XP, +10 coins

Linking sin, cos and tan

Show, using SOH-CAH-TOA, that $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$ for any acute $\theta$ in a right triangle.

Reveal solution

$\dfrac{\sin\theta}{\cos\theta} = \dfrac{\text{opp}/\text{hyp}}{\text{adj}/\text{hyp}} = \dfrac{\text{opp}}{\text{hyp}} \cdot \dfrac{\text{hyp}}{\text{adj}} = \dfrac{\text{opp}}{\text{adj}} = \tan\theta$. The hyp cancels.

Quick Review

SOH

$\sin\theta = $ Opp/Hyp

CAH

$\cos\theta = $ Adj/Hyp

TOA

$\tan\theta = $ Opp/Adj

Ratio constant

Same angle → same ratio

Tan special

Doesn't use the hypotenuse

Identity

$\tan\theta = \sin\theta / \cos\theta$

Your Badges

0 of 6

First Steps

3-Day Streak

3 in a Row

Lesson Ace

Stretch Seeker

Daily Warrior

Mark lesson as complete

Tick when you've finished Learn, Practice and the Stretch. Earns +85 XP and +25 coins.

Unit overview · Maths subject page