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Trigonometry

Identify the triangle and label known sides and angles before selecting a rule. Sine, cosine and tangent as side ratios require a right triangle. For a general triangle, use the sine or cosine rule when the given information fits.

What this guide covers

Pythagoras, right-triangle ratios, sine and cosine rules, triangle area and 2D/3D applications including bearings. The short checks sample right triangles and area, not the full topic.

Syllabus reference: 2026 Mathematics 4052, G4, page 9. These are Mentora search groupings, not new official syllabus categories.

The content in this guide was also checked against 2027 G3 Mathematics K310, G4, page 9 on 2026-09-12. This is subject-content alignment, not a claim of complete syllabus coverage.

Which rule fits the information?

Right angle and two sides
Use Pythagoras for a missing side. For an acute angle, choose sin, cos or tan from the labelled side pair.
A known opposite side–angle pair
Use a/sin A = b/sin B. With two angles, first find the third. When finding an angle from sine, check whether the supplementary angle also fits the given triangle.
Two sides and their included angle
Use c² = a² + b² − 2ab cos C for the third side. The angle C must lie between a and b.
Three sides
Use cos C = (a² + b² − c²)/(2ab). Pair C with the opposite side c.

Worked example

A right triangle has hypotenuse 14 cm and an acute angle of 30°. Find the side opposite that angle.

Use sin 30° = opposite/14. Hence opposite = 14 × sin 30° = 7 cm. The answer is less than the hypotenuse, as it should be.

Sine rule: match opposite pairs

In triangle ABC, A = 40°, B = 65° and a = BC = 7 cm. Find b = AC to 3 significant figures.

The known opposite pair is a = 7 and A = 40°. Hence b/sin 65° = 7/sin 40°, so b = 7 sin 65°/sin 40° ≈ 9.87 cm. Since B is larger than A, b should be longer than a. The remaining angle is 75°, so the angles form a valid triangle.

Cosine rule: the included angle

Two sides of a triangle measure 5 cm and 8 cm and enclose 60°. Find the third side c.

No opposite side–angle pair is known. Use c² = 5² + 8² − 2 × 5 × 8 × cos 60° = 49, so c = 7 cm exactly. Do not use the sine rule with 60° paired to 5 or 8; neither is opposite that angle.

Bearings: establish the triangle first

A boat travels from A, 6 km due north to B, then 8 km on a bearing of 090° to C. Find the distance and bearing of C from A.

ABCN6 km8 kmθ
A to B: 6 km north; B to C: 8 km east. The right angle is at B. θ at A is measured clockwise from north towards AC.

BC is due east, perpendicular to AB. AC = √(6² + 8²) = 10 km exactly. At A, the clockwise angle from north satisfies tan θ = 8/6, so θ ≈ 53.1°. The bearing is 053.1° (to 1 decimal place), not the 36.9° angle measured from east. The bearing of A from C would instead be 233.1°.

Try three practice checks

Attempt each on paper before revealing the worked answer. These sample particular skills; they do not diagnose the whole topic or establish mastery.

Check 1

A right triangle has perpendicular sides 9 cm and 12 cm. Find its hypotenuse.

Show worked answer

By Pythagoras, c² = 9² + 12² = 225, so c = 15 cm.

Check 2

In a right triangle, the opposite and adjacent sides to angle θ are 5 cm and 12 cm. Find θ to 1 decimal place.

Show worked answer

tan θ = 5/12. In degree mode, θ = tan⁻¹(5/12) ≈ 22.6°.

Check 3

Two sides of a triangle are 8 cm and 11 cm, with included angle 30°. Find its area.

Show worked answer

Area = ½ × 8 × 11 × sin 30° = 22 cm². The angle must be between the two stated sides.

Mistakes to watch for

  • Opposite and adjacent depend on the chosen angle; the hypotenuse is always opposite the right angle.
  • Use inverse trigonometry to find an angle and check degree mode when answering in degrees.
  • In 3D, identify the relevant right triangle and its plane before calculating; a line angle and a plane angle are different.

If one of these occurs, compare the relevant step with the worked example, then try a different question before drawing conclusions.

Useful foundations to check

These connections suggest what may help with this skill. Selecting a topic says nothing about your current attainment.

Applications and next topics

Use trigonometry for heights, bearings and triangle areas. Coordinate distance and vector magnitude also use Pythagoras.

Choose a useful next resource

To understand a recurring error rather than relearn the whole skill: Check trig rule and calculator-mode mistakes.

Explore the foundations connected to trigonometry

Sources and review

Original Mentora public practice, authored for this guide. Worked answers and syllabus mapping checked during editorial QA on 2026-09-12; no independent expert review is claimed. Graph connections are Mentora learning hints, not an official SEAB sequence. Read the source and examination-year notes.