Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 2 - Trigonometry, direction, elevation and depression

Trigonometry, direction, elevation and depression

Use right-triangle trigonometry to solve problems involving bearings, direction, elevation and depression.

Updated 2026-07-26 - 11 min read

Trigonometry, direction, elevation and depression is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Select sine, cosine or tangent

Relative to the reference angle, choose the ratio connecting the known and required sides: SOH, CAH or TOA. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by select sine, cosine or tangent and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Opposite and adjacent depend on the chosen reference angle; only the hypotenuse is fixed.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

Relative to θ, opposite=8 and hypotenuse=10. Which equation applies?

Step 1 - identify the governing idea: Relative to the reference angle, choose the ratio connecting the known and required sides: SOH, CAH or TOA.

Step 2 - apply it to this evidence: Sine connects opposite and hypotenuse.

Result: sin θ = 8/10

The relationship is visible in the working: Sine connects opposite and hypotenuse. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • cos θ = 8/10 — It does not agree with the required relationship: Sine connects opposite and hypotenuse.
  • tan θ = 10/8 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • sin θ = 10/8 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

Adjacent=12 and hypotenuse=13. Which ratio finds θ?

Step 1 - identify the governing idea: Relative to the reference angle, choose the ratio connecting the known and required sides: SOH, CAH or TOA.

Step 2 - apply it to this evidence: Cosine connects adjacent and hypotenuse.

Result: cos θ = 12/13

The relationship is visible in the working: Cosine connects adjacent and hypotenuse. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • sin θ = 12/13 — It does not agree with the required relationship: Cosine connects adjacent and hypotenuse.
  • tan θ = 13/12 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • cos θ = 13/12 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

Opposite=7 and adjacent=4. Which ratio finds θ?

Step 1 - identify the governing idea: Relative to the reference angle, choose the ratio connecting the known and required sides: SOH, CAH or TOA.

Step 2 - apply it to this evidence: Tangent connects opposite and adjacent.

Result: tan θ = 7/4

The relationship is visible in the working: Tangent connects opposite and adjacent. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • sin θ = 7/4 — It does not agree with the required relationship: Tangent connects opposite and adjacent.
  • cos θ = 4/7 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • tan θ = 4/7 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Solve right-triangle lengths and angles

Draw and label the triangle, form a ratio, use the appropriate direct or inverse function and check scale and angle range. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by solve right-triangle lengths and angles and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Inverse trigonometric functions find angles; ordinary ratios find side lengths.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

A right triangle has hypotenuse 10 and angle 30°. Find the opposite side.

Step 1 - identify the governing idea: Draw and label the triangle, form a ratio, use the appropriate direct or inverse function and check scale and angle range.

Step 2 - apply it to this evidence: Opposite = 10 sin 30° = 5.

Result: 5

The relationship is visible in the working: Opposite = 10 sin 30° = 5. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 8.66 — It does not agree with the required relationship: Opposite = 10 sin 30° = 5.
  • 20 — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0.5 — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

Opposite=6 and adjacent=8. Find θ to one decimal place.

Step 1 - identify the governing idea: Draw and label the triangle, form a ratio, use the appropriate direct or inverse function and check scale and angle range.

Step 2 - apply it to this evidence: θ = tan⁻¹(6/8) ≈ 36.87°.

Result: 36.9°

The relationship is visible in the working: θ = tan⁻¹(6/8) ≈ 36.87°. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 53.1° — It does not agree with the required relationship: θ = tan⁻¹(6/8) ≈ 36.87°.
  • 0.8° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 48.6° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

A 12 m ladder makes 65° with the ground. Find vertical reach to one decimal place.

Step 1 - identify the governing idea: Draw and label the triangle, form a ratio, use the appropriate direct or inverse function and check scale and angle range.

Step 2 - apply it to this evidence: Height = 12 sin 65° ≈ 10.88 m.

Result: 10.9 m

The relationship is visible in the working: Height = 12 sin 65° ≈ 10.88 m. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 5.1 m — It does not agree with the required relationship: Height = 12 sin 65° ≈ 10.88 m.
  • 25.7 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 11.2 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Model bearings, elevation and depression

Translate the wording into a horizontal-reference diagram, measure bearings clockwise from north and use equal alternate angles for horizontal sight lines. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by model bearings, elevation and depression and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Elevation and depression are measured from a horizontal line of sight.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

A point has bearing 135°. In which general direction is it?

Step 1 - identify the governing idea: Translate the wording into a horizontal-reference diagram, measure bearings clockwise from north and use equal alternate angles for horizontal sight lines.

Step 2 - apply it to this evidence: 135° is clockwise from north, midway between east and south.

Result: South-east

The relationship is visible in the working: 135° is clockwise from north, midway between east and south. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • North-west — It does not agree with the required relationship: 135° is clockwise from north, midway between east and south.
  • North-east — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • South-west — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

From 40 m away, the angle of elevation to a tower top is 32°. Find height to one decimal place.

Step 1 - identify the governing idea: Translate the wording into a horizontal-reference diagram, measure bearings clockwise from north and use equal alternate angles for horizontal sight lines.

Step 2 - apply it to this evidence: Height = 40 tan 32° ≈ 24.99 m.

Result: 25.0 m

The relationship is visible in the working: Height = 40 tan 32° ≈ 24.99 m. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 64.0 m — It does not agree with the required relationship: Height = 40 tan 32° ≈ 24.99 m.
  • 21.2 m — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 33.9 m — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

An observer sees a boat at 18° depression. What is the boat's angle of elevation to the observer?

Step 1 - identify the governing idea: Translate the wording into a horizontal-reference diagram, measure bearings clockwise from north and use equal alternate angles for horizontal sight lines.

Step 2 - apply it to this evidence: Parallel horizontal lines make the alternate interior angles equal.

Result: 18°

The relationship is visible in the working: Parallel horizontal lines make the alternate interior angles equal. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 72° — It does not agree with the required relationship: Parallel horizontal lines make the alternate interior angles equal.
  • 162° — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 0° — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. Adjacent=12 and hypotenuse=13. Which ratio finds θ?
  2. Opposite=6 and adjacent=8. Find θ to one decimal place.
  3. From 40 m away, the angle of elevation to a tower top is 32°. Find height to one decimal place.

Answers

  1. cos θ = 12/13 — Cosine connects adjacent and hypotenuse.
  2. 36.9° — θ = tan⁻¹(6/8) ≈ 36.87°.
  3. 25.0 m — Height = 40 tan 32° ≈ 24.99 m.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources