IGCSE Maths trigonometry: SOHCAHTOA, the sine rule and the cosine rule

What each tier expects

The key ideas

Label the sides of a right-angled triangle relative to the angle you are using: the hypotenuse is opposite the right angle, the opposite side faces your angle, and the adjacent side is next to it. Then choose the ratio that links the side you know with the side you want. To find an angle, use the inverse function, for example x = tan^-1(5/12).

For a non-right-angled triangle, label the sides a, b, c opposite the angles A, B, C. If the question gives you a matching pair (a side and the angle opposite it), use the sine rule. If it gives two sides and the included angle, or all three sides, use the cosine rule. Rearranged for an angle, the cosine rule is cos A = (b^2 + c^2 - a^2)/(2bc).

The sine rule for an angle can give two answers, because sin x = sin(180° - x). This is the ambiguous case: check whether the obtuse option still leaves room in the triangle. The area of any triangle is 1/2 ab sin C, using two sides and the angle between them.

Cambridge Extended expects exact values without a calculator: sin 30° = 1/2, cos 30° = sqrt(3)/2, sin 45° = cos 45° = 1/sqrt(2), tan 45° = 1, tan 60° = sqrt(3), plus solving equations such as 2cos x + 1 = 0 for 0° ≤ x ≤ 360° using the graphs.

Worked example 1: right-angled triangles

1. A 6 m ladder leans against a wall, making 70° with the ground. How high up the wall does it reach?
2. The height is opposite the 70° angle and the ladder is the hypotenuse, so use sin: height = 6 sin 70° = 5.638...
3. Answer: 5.64 m (3 s.f.).
4. Second part: a right-angled triangle has opposite 5 cm and adjacent 12 cm. The angle is tan^-1(5/12) = 22.619...°, so 22.6° to 1 d.p. Check the hypotenuse with Pythagoras: sqrt(25 + 144) = 13, and sin^-1(5/13) gives the same 22.6°.

Worked example 2: cosine rule and area

1. Triangle PQR has PQ = 7 cm, PR = 9 cm and angle QPR = 50°. Find QR and the area.
2. Two sides and the included angle, so use the cosine rule: QR^2 = 7^2 + 9^2 - 2 × 7 × 9 × cos 50° = 130 - 126 cos 50° = 130 - 80.99... = 49.008...
3. QR = sqrt(49.008...) = 7.0006..., so QR = 7.00 cm (3 s.f.). Keep the full value in the calculator for later parts.
4. Area = 1/2 × 7 × 9 × sin 50° = 31.5 × 0.7660... = 24.13..., so 24.1 cm^2.

Worked example 3: sine rule and the ambiguous case

1. In triangle ABC, a = 8 cm, b = 11 cm and angle A = 40°. Find the possible values of angle B.
2. Matching pair a and A, so use the sine rule: sin B / 11 = sin 40° / 8, so sin B = 11 sin 40° / 8 = 0.8838...
3. B = sin^-1(0.8838...) = 62.1°, or B = 180° - 62.1° = 117.9°.
4. Check the obtuse option: 40° + 117.9° = 157.9°, which is less than 180°, so both triangles exist. Answer: B = 62.1° or 117.9°.

Common mistakes that cost marks

• Calculator in radian mode. Check that it shows D or DEG before the exam.
• Labelling opposite and adjacent from the wrong angle.
• Using the cosine rule as (b^2 + c^2 - 2bc) cos A, i.e. multiplying before subtracting in the wrong order. Work out 2bc cos A first, then subtract.
• Rounding QR to 7 and then using 7 in a later part, which shifts later answers.
• Missing the second angle in the ambiguous case, or giving it when the triangle cannot exist.
• Giving angles to 3 significant figures when Cambridge asks for 1 decimal place: 117.9°, not 118°.

Exam technique and how a tutor helps

Draw and label the triangle every time, even if a diagram is given, and mark what you know and what you want. That one step usually tells you which rule to use. In multi-step problems, such as bearings or 3D shapes, split the shape into separate triangles and solve them one at a time. For 3D questions, the angle between a line and a plane is found in the right-angled triangle formed by the line, its projection on the plane and the perpendicular.

Cambridge requires non-exact answers to 3 significant figures and angles to 1 decimal place, so carry full calculator values through and round only at the end.

Trigonometry is a topic where students often know the formulas but cannot decide which one to use. A tutor working one to one will ask the student to say out loud what they have and what they need before writing anything, which builds the decision habit, and then gives mixed sets where right-angled and non-right-angled triangles are deliberately interleaved.

Self-check: can you do these?

• A right-angled triangle has hypotenuse 10 cm and an angle of 35°. Find the side adjacent to the angle. (Answer: 8.19 cm)
• A triangle has sides 5, 6 and 7 cm. Find its largest angle. (Answer: 78.5°)
• Find the area of a triangle with sides 5 cm and 8 cm and an included angle of 30°. (Answer: 10 cm^2)
• From the top of a 12 m building, the angle of depression of a car is 25°. How far is the car from the base? (Answer: 25.7 m)
• Solve 2cos x + 1 = 0 for 0° ≤ x ≤ 360°. (Answer: 120° and 240°)