Math // Geometry

Trigonometry basics

Use the trigonometric ratios sin, cos, and tan in right-angled triangles to find unknown sides and angles, including setting up the correct ratio for a given problem

Ages
13 to 15
Type
Method
Builds on
2 topics
Leads to
0 topics
Standards
1 code

Quick check

If the learner knows a ladder is 5 metres long and leans against a wall at 70° to the ground, can they use trigonometry to work out how high up the wall it reaches?

Ask this to see whether the topic is secure.

Topic // Mastery

What mastery looks like

What a learner can do once this topic is secure.
  • Identify which trigonometric ratio to use (SOH CAH TOA) based on the known and unknown sides
  • Calculate an unknown side using sin, cos, or tan and a known angle
  • Find an unknown angle using the inverse trigonometric function (e.g. tan⁻¹)

Topic // Map

Where this fits

What to learn first and what this topic opens up, with the reason for each link. Required links come first.
  • RequiredMust come first
  • HelpfulHelps, but not essential

Before this

2 topics
  • Simple formulaeRequired

    Algebra // Ages 12 to 14

    Using trig ratios to find unknowns requires formula rearrangement

  • Applying Pythagoras' theoremRequired

    Geometry // Ages 13 to 14

    Trigonometry extends Pythagoras by introducing angle-side relationships

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Trigonometry basics

Ages 13 to 15 // Method

Leads to

0 topics

Nothing in the catalogue builds on this topic yet. Browse Geometry

Topic // Research-informed

Teaching notes

From the Marble Skill Taxonomy. These notes have not been validated in classrooms, so treat them as a starting point for your own judgement.

The approach

Right-angled trigonometry (sin, cos, tan) relates a reference angle θ to a pair of sides. The two shorter sides are named RELATIVE TO θ — opposite is across from θ, adjacent is next to θ — while only the hypotenuse is fixed (opposite the right angle, longest side). The central danger is reducing trigonometry to mnemonic recitation (SOHCAHTOA) without understanding, so learners mislabel sides, pick the wrong ratio and cannot recover when a triangle is rotated or when they work from the other acute angle. Ground the ratios in similarity/ratio-invariance: for a fixed angle the within-triangle ratio (e.g. opposite÷hypotenuse) is constant across similar triangles, and that constant IS sin of the angle. Distinguish finding an unknown SIDE (rearrange the ratio) from finding an unknown ANGLE (apply the inverse function).

Common misconceptions

  • Labels opposite and adjacent relative to the RIGHT angle instead of the reference angle θ, so every ratio is set up wrong
  • Relies on the SOHCAHTOA mnemonic without understanding, and cannot choose which ratio applies when given a particular pair of sides
  • Mislabels the sides when the triangle is drawn rotated or in a non-standard orientation
  • For an unknown side, cannot rearrange the ratio to isolate it (e.g. leaves the unknown in the denominator)
  • For an unknown angle, forgets to use the inverse function (uses sin instead of sin⁻¹) or reports the ratio value as if it were the angle

Useful representations

  • A right triangle with a movable reference-angle marker θ, and opposite/adjacent labels that re-attach as θ moves
  • The unit circle with the rotating radius, showing sinθ as the y-coordinate and cosθ as the x-coordinate
  • Nested similar right triangles showing opposite÷hypotenuse constant for a fixed angle (ratio invariance)
  • SOH-CAH-TOA presented as three side-pairs, each ratio highlighting exactly the two sides it uses

Teaching practices

  • Mark the reference angle θ FIRST, then label opposite (across from θ) and adjacent (next to θ, and NOT the hypotenuse) relative to it — never relative to the right angle
  • Show explicitly that opposite and adjacent SWAP when you switch to the other acute angle, while the hypotenuse stays put — so the labels depend on which angle you are working from
  • Ground the ratios in similarity/ratio-invariance before the mnemonic: for a fixed angle, opposite÷hypotenuse is the same across nested similar triangles (NCETM's unit-circle and scale-factor route), so the ratio has meaning, not just letters
  • Sequence problem-solving: identify the two sides involved (one known, one unknown), then choose the single ratio that contains exactly those two sides
  • Distinguish the two problem types — unknown SIDE (rearrange the ratio, e.g. o = h·sinθ) vs unknown ANGLE (apply the inverse function, e.g. θ = tan⁻¹(o/a))
  • Start in a standard orientation to keep the unit-circle link visible, then deliberately rotate triangles once the naming convention is secure
  • Reasonableness checks: the hypotenuse is the longest side, and an acute angle in a right triangle is between 0° and 90°

Activities to try

  1. 1

    Labels that move with the reference angle

    Why it works: Opposite and adjacent are not glued to the shape; they depend on which angle you are working from. Only the hypotenuse is fixed by the right angle.

    1. Mark the reference angle θ FIRST on a right-angled triangle, then label 'opposite' (across from θ) and 'adjacent' (next to θ, and NOT the hypotenuse) relative to it.
    2. Label the hypotenuse as the side opposite the right angle — the longest side, which stays fixed.
    3. Now mark the OTHER acute angle as θ on the same triangle and relabel: notice opposite and adjacent SWAP places while the hypotenuse stays put.
    4. Repeat on a triangle drawn rotated out of standard orientation, labelling relative to θ rather than to the shape.
    5. If a draggable θ marker is available, drag θ between the two acute vertices and watch the opposite/adjacent labels re-attach and swap live while the hypotenuse label stays fixed.
  2. 2

    The ratio that refuses to change (similarity before the mnemonic)

    Why it works: For a given angle the ratio of two sides is invariant regardless of the triangle's size, and that constant value IS sin/cos/tan of the angle — so the ratio has meaning, not just letters.

    1. Keep the angle θ fixed and scale a right-angled triangle bigger and smaller (nested similar triangles).
    2. For each size, measure or read opposite and hypotenuse and compute opposite÷hypotenuse; record the values in a table.
    3. Confirm the ratio stays constant (for example 0.42 at every size when θ = 25°) even though the side lengths all change.
    4. Connect this constant to a unit-circle inset: it is named sin θ (with sin θ as the y-coordinate and cos θ as the x-coordinate of the rotating radius).
    5. If a live-scaling model is available, drag to resize the triangle and watch a readout of opp÷hyp stay fixed while the lengths change, with the unit-circle inset naming the constant.
  3. 3

    Choose the ratio, then solve for a side or an angle

    Why it works: Ratio selection follows from the pair of sides involved; and whether you rearrange or invert depends on whether the unknown is a side or an angle.

    1. Given a problem, identify the two sides involved — one known, one unknown — relative to the marked angle θ.
    2. Choose the single ratio (SOH, CAH or TOA) that contains exactly those two sides.
    3. For an unknown SIDE: rearrange the ratio to make the unknown the subject, e.g. from sin 35° = o/10 derive o = 10 × sin 35° ≈ 5.74 cm. (Check: the opposite side cannot exceed the hypotenuse.)
    4. For an unknown ANGLE: apply the inverse function, e.g. from tan θ = 6/8 (opposite 6, adjacent 8) derive θ = tan⁻¹(0.75) ≈ 36.9°. (Check: an acute angle lies between 0° and 90°.)
    5. Practise a worked case where adjacent and hypotenuse are known: use cosine, θ = cos⁻¹(adjacent/hypotenuse).

Topic // Standards

Curriculum standards

Codes for the curriculum statements this topic maps to. Each publisher owns its standard text.
The national curriculum in England: Key stages 1 and 2 framework document
KS3.Maths.Geo.14

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