Year 10 Mathematics · AC9M10M03

Pythagoras and Right-Triangle Trigonometry

Draw and label the right triangle first, then choose Pythagoras or the trigonometric ratio that links the known and unknown quantities

Ready to project and teach

Learning goalsSay it simply

explain the central idea: Use Pythagoras and right-triangle trigonometry to solve practical direction, bearing, elevation and depression problems.
choose and apply an appropriate method without relying on keyword matching
check results using units, substitution, estimation, a second representation or contextual reasonableness
justify a conclusion and communicate limitations where the context requires them

Key conceptTeach from the board

Pythagoras applies to side lengths in right triangles. Sine, cosine and tangent connect an acute angle with side ratios. Bearings are measured clockwise from north; angles of elevation/depression are measured from a horizontal line of sight.

Pythagoras

c²=a²+b², with c opposite the right angle.

SOH–CAH–TOA

sinθ=opp/hyp

cosθ=adj/hyp

tanθ=opp/adj

Angles

θ=sin⁻¹(opp/hyp), cos⁻¹(adj/hyp), tan⁻¹(opp/adj).

Bearings

Three-digit, clockwise from north: 045°, 120°, 275°.

Worked examplesWe do

Worked examples

AC9M10M03 - Pythagoras and Right-Triangle Trigonometry

Example 1

Pythagoras c²=a²+b², with c opposite the right angle.

Example 2

SOH–CAH–TOA sinθ=opp/hyp cosθ=adj/hyp tanθ=opp/adj

Example 3

Angles θ=sin⁻¹(opp/hyp), cos⁻¹(adj/hyp), tan⁻¹(opp/adj).

Example 4

Bearings Three-digit, clockwise from north: 045°, 120°, 275°.

Curriculum examplesCopied content

Australian Curriculum: AC9M10M03 — Year 10 Measurement. Use Pythagoras and right-triangle trigonometry to solve practical direction, bearing, elevation and depression problems.

Victoria: VC2M10M03 — Level 10 Measurement

NSW: Stage 5 Core — Pythagoras and trigonometry; Path — Further trigonometry

Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.

Lesson component	Australian Curriculum	Victoria	NSW
Explicit concept teaching and worked examples	AC9M10M03	VC2M10M03 — Level 10 Measurement	Stage 5 Core — Pythagoras and trigonometry; Path — Further trigonometry
Guided and independent practice	Builds fluency and application for AC9M10M03	Practises the mapped Level 10/10A knowledge as applicable	Practises the mapped Stage 5 Core/Path content
Reasoning and assessment tasks	Applies reasoning/problem solving in the descriptor context	Supports Victorian reasoning and modelling expectations	Embeds Working mathematically: reasoning, problem solving and communication

Australian Curriculum elaborations

AC9M10M03: solve practical problems applying Pythagoras’ theorem and trigonometry of right-angled triangles, including direction and angles of elevation/depression.

E1: navigation/bearings, such as an orienteering final leg. Examples 5–6.
E2: decompose 3D surveying/design problems into 2D right triangles, such as smallest packaging. Example 9.
E3: use a clinometer and proportional/trig reasoning to find building heights. Example 7.
E4: model elevation/depression in practical 3D situations. Example 8.
E5: investigate autonomous-vehicle distance/navigation algorithms using coordinate distance. Example 10.
E6: explore First Nations Australian navigation, surveying or technology design and connections to geometric/spatial reasoning. Use authentic community/source material rather than generic attribution.

Questions and answersWith answers

Find hypotenuse for legs 9 and 12.: 15.
Find missing leg when c=17 and other leg=8.: 15.
Find opposite if θ=28°, adjacent=14.: 14tan28°≈7.44.

Practice and reviewReady for practice

Using Pythagoras on a triangle that is not right-angled.
Choosing opposite/adjacent before identifying the reference angle.
Calculator in radians instead of degrees.
Measuring bearings from east or anticlockwise.
Forgetting observer/eye height in elevation problems.

Classroom View Homework Practice Test

Curriculum alignmentStart here

explain the central idea: Use Pythagoras and right-triangle trigonometry to solve practical direction, bearing, elevation and depression problems.
choose and apply an appropriate method without relying on keyword matching
check results using units, substitution, estimation, a second representation or contextual reasonableness
justify a conclusion and communicate limitations where the context requires them

Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.

Other curriculum comparisons retained

Closest US CCSS alignment HSG-SRT.C.8 and coordinate geometry; UK GCSE Pythagoras/trigonometry/bearings; NSW Stage 5 and Victorian Level 10 Measurement/Space; comparable Canadian/NZ/CBSE trigonometry.

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