Lesson 20 ~25 min Unit 3 · Trigonometry +85 XP

Trigonometry Synthesis and Review

Bring it all together: SOH-CAH-TOA, Pythagoras, inverse trig, elevation/depression, and bearings. Mixed exam-style questions to consolidate everything.

Today's hook: From triangles to towers to navigation — trig is everywhere. This review tests every skill from the unit. Prove your mastery.

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

Look back on Lessons 1-19: which topic felt easiest? Which felt hardest? For your hardest topic, write ONE rule you'll remember to use it correctly.

Record your answer in your workbook.

The Big Idea

+5 XP

You've now seen all the tools of Year 9 trigonometry. Your job in this lesson is to identify which tool fits each problem, then apply it.

Pythagoras for unknown sides in a right triangle (you know the other two). Sin/cos/tan when you know an angle and a side, and want another side. $\sin^{-1}/\cos^{-1}/\tan^{-1}$ when you know two sides and want an angle. Elevation/depression for looking up/down problems. Bearings for navigation. Multi-step when no single tool suffices.

Six tools, one question: which fits THIS problem?

What's known?

List everything given in the problem before picking a tool.

What's asked?

Underline the unknown so you don't accidentally solve a different question.

Sketch

Every problem deserves a small diagram — saves choosing the wrong ratio.

What You'll Master

objectives

Know

The six main tools of Year 9 trigonometry
Which tool applies to which problem type
Standard rounding conventions: 2 d.p. for distances, 1 d.p. for angles

Understand

How tools combine in multi-step problems
Why one tool is needed for each unknown found
How to verify answers via Pythagoras or angle sum

Can Do

Solve mixed-style trig problems independently
Choose the correct tool and rearrangement
Communicate solutions clearly with intermediate steps

Words You Need

vocabulary

SOH-CAH-TOASin = Opp/Hyp, Cos = Adj/Hyp, Tan = Opp/Adj.

Pythagoras$a^2 + b^2 = c^2$ in a right triangle.

Inverse trig$\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ — find an angle from two sides.

Angle of elevationFrom horizontal looking up.

Angle of depressionFrom horizontal looking down.

Compass / true bearing$N\theta E$ vs three-digit clockwise from N.

Spot the Trap

heads-up

Wrong: Picking sin/cos/tan based on which is ‘easier’ instead of which fits the side pair.

Right: Pick the ratio based on the two sides involved: opp+hyp → sin; adj+hyp → cos; opp+adj → tan.

Wrong: Rounding mid-problem — especially in multi-step.

Right: Round only at the END; keep full calculator precision through intermediate steps.

The Decision Tree

+5 XP

Use this tree to pick your tool for any trig question:

What's given?	What's wanted?	Tool
2 sides of right triangle	3rd side	Pythagoras
Angle + 1 side	Another side	sin / cos / tan
2 sides	An angle	Inverse trig
Bearing + distance	N-E components	$d\cos$, $d\sin$
Need intermediate length	Final answer	Multi-step (combine)

Identify given + wanted, then pick tool

Slow is fast

A 20-second sketch saves 5 minutes of confusion.

Tool first

Pick the right tool before touching the calculator.

Verify

After computing, check your answer makes physical sense.

Quick Formula Sheet

+5 XP

Everything in one place:

Formula	Use
$a^2 + b^2 = c^2$	Right triangle sides
$\sin\theta = $ opp/hyp	Find side or angle
$\cos\theta = $ adj/hyp	Find side or angle
$\tan\theta = $ opp/adj	No hyp involved
$\theta + \phi = 90°$	Two acute angles
Reverse bearing $\pm 180°$	Return journey

All formulas in one place — pick the one that fits

Memorise the three trig ratios

SOH-CAH-TOA is non-negotiable.

Pythagoras everywhere

Right triangles — always recognise them.

Calculator: DEG

Check at the start of every problem.

Watch Me Solve It · 3 examples

Watch Me Solve It · Mixed: angle + ramp

+15 XP per step

PROBLEM

A ramp 10 m long rises at 18° to the horizontal. Find (a) the vertical rise and (b) the horizontal run (2 d.p.).

1

Identify

hyp = 10, $\theta = 18°$. Want opp and adj.
2

Opp (rise)

opp $= 10\sin 18° \approx 3.09$ m
3

Adj (run)

adj $= 10\cos 18° \approx 9.51$ m

AnswerRise $\approx 3.09$ m; Run $\approx 9.51$ m

Watch Me Solve It · Mixed: tower + bearing

+15 XP per step

PROBLEM

From a 50 m tower top, a boat on bearing 120° is at depression 25°. Find the horizontal distance from tower to boat (2 d.p.), and the boat's east-west displacement from the tower.

1

Horizontal distance

$d = 50/\tan 25° \approx 107.24$ m
2

East component

$E = d\sin 120° \approx 107.24 \times 0.866 \approx 92.86$ m
3

Conclude

Boat is about 92.86 m east of tower.

Answer107.24 m horizontal; 92.86 m east

Watch Me Solve It · Multi-step: cliff + walk + return

+15 XP per step

PROBLEM

A walker is at the foot of a cliff. They walk 80 m on bearing 045°, then turn and look back. The cliff-top is now at elevation 12°. Find the cliff height (2 d.p.).

1

Horizontal distance from cliff

After walking 80 m, the horizontal distance from cliff base is 80 m.
2

Use elevation

$h = 80\tan 12° \approx 17.00$ m
3

Cliff height

About 17.00 m.

Answer$\approx 17.00$ m

Common Pitfalls

heads-up

Picking the wrong tool

Using a side problem's method on an angle problem.

Fix: Identify given + wanted FIRST.

Rounding too early in multi-step

Accumulating errors over multiple stages.

Fix: Keep full precision; round only the final answer.

Calculator in wrong mode

RAD instead of DEG.

Fix: Verify $\sin 30° = 0.5$ at the start of every session.

Copy Into Your Books

Six tools

Pythagoras
Trig ratios
Inverse trig
Elev/Depres
Bearings
Multi-step

Decision

Sketch the problem
List given/wanted
Pick tool
Compute carefully

Precision

Full through steps
Round at end
Add units

Verify

Sense check
Angle sum
Pythagoras

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Synthesis

4 problems

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

1 Right triangle, hyp = 10, $\theta = 30°$. Opp?

$10\sin 30 = 5$.5
2 Right triangle, legs 5 and 12. Third side?

$\sqrt{25+144}=13$.13
3 Right triangle, opp = 4, hyp = 5. Angle?

$\sin^{-1}(4/5) \approx 53.1°$.$\approx 53.1°$
4 Bearing 060° for 8 km. N-component (2 d.p.)?

$8\cos 60° = 4$.4 km N

Complete in your workbook.

Quick Check · 5 questions

In a right triangle, you know hyp and want opp. Use:

+10 XP

A right triangle has legs 7 and 24. Hyp:

+10 XP

From 40 m, a tower is at elevation 38°. Height (2 d.p.):

+10 XP

True bearing 245° expressed in compass:

+10 XP

opp = 3, adj = 4. Angle $\theta$:

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

ApplyMedium3 MARKS

Q6. A right triangle has hypotenuse 26 cm and one leg 10 cm. (a) Find the other leg. (b) Find both acute angles (1 d.p.).

Answer in your workbook.

ApplyMedium3 MARKS

Q7. A surveyor on flat ground measures the elevation of a mountain peak as 12° from point A. Moving 200 m closer to the mountain on the same line, the elevation from point B is 18°. Find the height of the mountain (2 d.p.).

Answer in your workbook.

ReasonHard4 MARKS

Q8. A ship sails 30 km on bearing 080°, then 20 km on bearing 200°. (a) Find the ship's position relative to start (N/E components, 2 d.p.). (b) Find the straight-line distance from start. (c) Find the true bearing on which the ship should sail to return directly to start.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. B — sin.

2. D — 7-24-25 triple.

3. C — $40\tan 38° \approx 31.25$.

4. A — $S65°W$.

5. B — $\tan^{-1}(0.75) \approx 36.9°$.

Show Your Working Model Answers

Q6 (3 marks): (a) Other leg $= \sqrt{676 - 100} = \sqrt{576} = 24$ cm (10-24-26 = $2\times$(5-12-13)) [1]. (b) Angle opposite the 10: $\sin^{-1}(10/26) \approx 22.6°$ [1]. Other acute: $90° - 22.6° = 67.4°$ [1].

Q7 (3 marks): Let $d$ = distance from B. $h = d\tan 18°$ and $h = (d+200)\tan 12°$ [1]. Solve: $d(\tan 18° - \tan 12°) = 200\tan 12°$. $d \approx 42.51/0.1124 \approx 378.18$ m [1]. $h \approx 378.18\tan 18° \approx 122.88$ m [1].

Q8 (4 marks): (a) Leg 1: N $= 30\cos 80° \approx 5.21$, E $= 30\sin 80° \approx 29.54$. Leg 2: N $= 20\cos 200° \approx -18.79$, E $= 20\sin 200° \approx -6.84$ [1]. Net: N $\approx -13.58$, E $\approx 22.70$ [1]. (b) Distance $= \sqrt{13.58^2 + 22.70^2} \approx 26.45$ km [1]. (c) Bearing from start to ship: SE quadrant (N negative, E positive). Angle from S = $\tan^{-1}(22.70/13.58) \approx 59.1°$. True bearing $\approx 180 - 59.1 = 120.9°$. Return bearing $= 120.9 + 180 = 300.9°$ — round to 301° [1].

Stretch Challenge · +25 XP, +10 coins

Final master problem

A drone takes off from base camp B. It flies 600 m on bearing 050° while climbing to an altitude of 80 m. It then continues on bearing 130° for 400 m, descending to 30 m altitude. (a) Find the drone's position relative to B (N/E/altitude). (b) Find the straight-line 3D distance from B to the drone.

Reveal solution

Leg 1: N $= 600\cos 50 \approx 385.67$, E $= 600\sin 50 \approx 459.63$, altitude $+80$. Leg 2: N $= 400\cos 130 \approx -257.12$, E $= 400\sin 130 \approx 306.42$, altitude $-50$ (from 80 to 30, so change of $-50$). Net: N $\approx 128.55$, E $\approx 766.05$, altitude $30$ above B. 3D distance $= \sqrt{128.55^2 + 766.05^2 + 30^2} \approx 777.4$ m.

Quick Review

Pythagoras

$a^2 + b^2 = c^2$

SOH-CAH-TOA

sin, cos, tan ratios

Inverse trig

$\sin^{-1}, \cos^{-1}, \tan^{-1}$

Elevation/depression

From horizontal up/down

Bearings

Compass & true

Multi-step

Combine tools

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