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

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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

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

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