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Trigonometry

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Trigonometry

Summary: Covers right-angled triangle trigonometry (SOHCAHTOA), exact trigonometric values for key angles, the sine and cosine rules for non-right-angled triangles, area of a triangle (1/2 ab sin C), 3D trigonometry, and bearings. Trigonometry is a central topic in IGCSE Mathematics 0580, examined in both Core (right-angled only) and Extended (all rules) tiers. Tags: igcse mathematics trigonometry Created: 2026-07-16 Last Updated: 2026-07-16

1. Right-Angled Triangle Trigonometry (SOHCAHTOA)

For a right-angled triangle, with angle θ:

  • sin θ = opposite / hypotenuse
  • cos θ = adjacent / hypotenuse
  • tan θ = opposite / adjacent

Finding a side: Choose the correct ratio, substitute the known values, solve.

Example: In a right-angled triangle, the hypotenuse is 12 cm and angle = 35°. Find the opposite side. sin 35° = opp / 12 → opp = 12 × sin 35° ≈ 6.88 cm.

Finding an angle: Use the inverse function (sin⁻¹, cos⁻¹, tan⁻¹).

Example: In a right-angled triangle, opposite = 5 cm, adjacent = 7 cm. Find angle θ. tan θ = 5/7 → θ = tan⁻¹(5/7) ≈ 35.5°.

2. Exact Trigonometric Values

These are tested without calculator access. The table must be memorised.

Angle θsin θcos θtan θ
0°010
30°1/2√3/21/√3 = √3/3
45°1/√2 = √2/21/√2 = √2/21
60°√3/21/2√3
90°10undefined

Memory aid: For sine, the values for 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2. For cosine, reverse the sine sequence.

Example: Without a calculator, evaluate sin 30° + cos 60° = 1/2 + 1/2 = 1.

Example: Without a calculator, evaluate tan 45° × sin 60° = 1 × √3/2 = √3/2.

3. Sine Rule

Used for non-right-angled triangles. Labelling convention: side a is opposite angle A, side b opposite angle B, side c opposite angle C.

a / sin A = b / sin B = c / sin C (equivalently: sin A / a = sin B / b = sin C / c)

When to use:

  • Two angles and one side (AAS or ASA) — find a missing side
  • Two sides and a non-included angle (SSA) — find a missing angle. Watch for the ambiguous case.

Ambiguous case (Extended): When given two sides and a non-included angle (SSA), there may be two possible triangles — the angle could be acute or obtuse. Check: if the side opposite the given angle is shorter than the other given side, and the calculated angle is acute, then 180° minus the acute angle may also be valid (provided it does not cause the angle sum to exceed 180°).

Example (finding a side): In triangle ABC, A = 40°, B = 75°, a = 8 cm. Find side b. b / sin 75° = 8 / sin 40° → b = 8 × sin 75° / sin 40° ≈ 8 × 0.9659 / 0.6428 ≈ 12.0 cm.

Example (finding an angle): In triangle ABC, a = 9, b = 12, A = 45°. Find angle B. sin B / 12 = sin 45° / 9 → sin B = 12 × sin 45° / 9 ≈ 0.9428. B₁ = sin⁻¹(0.9428) ≈ 70.5°. B₂ = 180° − 70.5° = 109.5°. Both are valid since A + B₂ = 154.5° < 180°.

4. Cosine Rule

Finding a side: a² = b² + c² − 2bc cos A

Finding an angle (rearranged): cos A = (b² + c² − a²) / (2bc)

When to use:

  • Three sides (SSS) — find any angle
  • Two sides and the included angle (SAS) — find the third side

Example (finding a side): In triangle ABC, b = 7, c = 9, A = 60°. Find a. a² = 7² + 9² − 2 × 7 × 9 × cos 60° = 49 + 81 − 126 × 0.5 = 130 − 63 = 67. a = √67 ≈ 8.19 cm.

Example (finding an angle): In triangle ABC, a = 5, b = 6, c = 7. Find angle A. cos A = (6² + 7² − 5²) / (2 × 6 × 7) = (36 + 49 − 25) / 84 = 60/84 = 5/7. A = cos⁻¹(5/7) ≈ 44.4°.

5. Area of a Triangle

Right-angled triangle: Area = 1/2 × base × perpendicular height

Any triangle: Area = 1/2 ab sin C (half the product of two sides and the sine of the included angle)

Heron’s formula (Extended): Area = √[s(s − a)(s − b)(s − c)], where s = (a + b + c) / 2

Example: Find the area of triangle ABC where a = 8 cm, b = 6 cm, C = 30°. Area = 1/2 × 8 × 6 × sin 30° = 24 × 0.5 = 12 cm².

6. 3D Trigonometry (Extended)

Angle between a line and a plane: The angle between the line and its projection onto the plane. Use right-angled triangle trigonometry.

Angle between two planes: Find the line of intersection, then lines in each plane perpendicular to that intersection. The angle between those two perpendiculars is the angle between the planes.

Strategy:

  1. Draw a clear, well-labelled diagram.
  2. Identify the right-angled triangle containing the required angle.
  3. Use Pythagoras’ theorem to find any unknown lengths in 3D.
  4. Use SOHCAHTOA to find the required angle.

7. Bearings

Bearings are measured:

  • Clockwise from North
  • Always given as three digits: 045°, 120°, 315°
  • They measure the direction from one point to another.

Bearings and trigonometry: Many bearing problems involve constructing a triangle (often right-angled) between points, then using SOHCAHTOA or the sine/cosine rules to find unknown distances or bearings.

Example: A ship sails 8 km due East then 6 km due North. Find its bearing from the starting point. Bearing = tan⁻¹(8/6) from North towards East = tan⁻¹(4/3) ≈ 053.1°. Written as 053° (three digits).

Sources

  • BBC Bitesize GCSE Mathematics — Trigonometric Ratios and Rules, BBC (free educational resource)
  • OpenStax Math — Trigonometry, Rice University (free, CC BY 4.0)
  • Cambridge IGCSE Mathematics 0580 — Trigonometry, Cambridge Assessment International Education
  • CK-12 Mathematics — Trigonometry, CK-12 Foundation (free, CC BY-NC 3.0)

Common Misconceptions

Students often think…But the correct understanding is…
”SOHCAHTOA works for any triangle”SOHCAHTOA only applies to right-angled triangles. Use sine/cosine rules for others.
”The sine rule always gives one answer for an angle”In the SSA case, there may be two possible angles (ambiguous case). Always check.
”The cosine rule is only for finding sides”The rearranged form cos A = (b² + c² − a²)/(2bc) finds angles when all three sides are known.
”Bearings are measured anticlockwise”Bearings are always clockwise from North.
”sin θ = opposite/hypotenuse in any orientation”The opposite and adjacent sides are relative to the specific angle being used. Re-label for each angle.
”tan 90° = infinity (or a very large number)“tan 90° is undefined (division by zero).

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