Trigonometry for Edexcel IGCSE Mathematics Specification B: Methods and Worked Examples

Trigonometry occupies a central position in the Edexcel IGCSE Mathematics Specification B specification. The topic is examined heavily and carries significant marks across both papers, requiring fluency in both right-angled triangle methods and the more general sine and cosine rules.

The edexcel igcse mathematics specification b trigonometry content falls under a single heading, "Trigonometric ratios and applications," but the scope is substantial. It spans SOHCAHTOA in right-angled triangles, the sine and cosine rules for any triangle, the area formula using sine, angles of elevation and depression, and bearings. Trigonometry questions frequently combine with Pythagoras, mensuration, and coordinate geometry, making this one of the most interconnected topics in the specification.

Trigonometric ratios in right-angled triangles

SOHCAHTOA

The three primary trigonometric ratios relate the sides of a right-angled triangle to its angles:

Ratio	Definition
sin(theta) = opposite / hypotenuse	SOH
cos(theta) = adjacent / hypotenuse	CAH
tan(theta) = opposite / adjacent	TOA

Worked example: In a right-angled triangle, the side opposite angle A is 7 cm and the hypotenuse is 15 cm. Find angle A.

sin(A) = 7/15 = 0.4667...

A = arcsin(0.4667) = 27.8 degrees (to 3 s.f.).

Worked example: A ladder of length 6 m leans against a wall, making an angle of 72 degrees with the ground. How high up the wall does the ladder reach?

The height is the side opposite the 72-degree angle; the ladder is the hypotenuse.

height = 6 x sin(72) = 6 x 0.9511 = 5.71 m (to 3 s.f.).

Identifying opposite, adjacent, and hypotenuse correctly is the foundation of every trigonometry question. The hypotenuse is always the longest side, opposite the right angle. The opposite and adjacent sides change depending on which angle you are working with. If you label them incorrectly, every calculation that follows will be wrong, even if your arithmetic is flawless.

Sine, cosine, and tangent of angles up to 180 degrees

The specification requires use of trigonometric ratios for angles up to 180 degrees, not just acute angles. The key identities are:

sin(180 - theta) = sin(theta). For example, sin(150) = sin(30) = 0.5.
cos(180 - theta) = -cos(theta). For example, cos(120) = -cos(60) = -0.5.
tan(180 - theta) = -tan(theta).

This matters when the sine rule produces an obtuse angle. If sin(A) = 0.6, then A could be arcsin(0.6) = 36.9 degrees or A = 180 - 36.9 = 143.1 degrees. The context of the problem determines which solution is valid.

The sine rule

The sine rule applies to any triangle (not just right-angled ones):

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

where a is the side opposite angle A, and so on. The trigonometry edexcel igcse exam uses this rule when you know either two angles and a side, or two sides and an angle opposite one of them.

Worked example: In triangle PQR, angle P = 42 degrees, angle Q = 73 degrees, and side p (opposite angle P) = 8 cm. Find side q (opposite angle Q).

Angle R = 180 - 42 - 73 = 65 degrees.

Using the sine rule: q/sin(73) = 8/sin(42).

q = 8 x sin(73)/sin(42) = 8 x 0.9563/0.6691 = 11.43 cm (to 4 s.f.).

The ambiguous case

When given two sides and a non-included angle, the sine rule can yield two possible triangles. If sin(B) = 0.8, then B = 53.1 degrees or B = 126.9 degrees. You must check whether both values produce a valid triangle (all angles positive and summing to 180 degrees).

The cosine rule

The cosine rule is used when you know two sides and the included angle, or all three sides:

a² = b² + c² - 2bc cos(A)

Rearranged to find an angle: cos(A) = (b² + c² - a²) / (2bc).

Worked example: In triangle ABC, b = 9 cm, c = 12 cm, and angle A = 55 degrees. Find side a.

a² = 81 + 144 - 2(9)(12)cos(55).

a² = 225 - 216 x 0.5736 = 225 - 123.9 = 101.1.

a = sqrt(101.1) = 10.05 cm (to 4 s.f.).

Worked example: In triangle DEF, d = 7, e = 10, f = 13. Find angle F.

cos(F) = (49 + 100 - 169) / (2 x 7 x 10) = -20/140 = -1/7.

F = arccos(-1/7) = 98.2 degrees (to 3 s.f.).

The negative cosine confirms that angle F is obtuse, which is consistent with side f being the longest side.

Choosing between the sine rule and the cosine rule. Use the sine rule when you have a known angle-side pair (an angle and the side opposite it). Use the cosine rule when you have two sides and the included angle, or all three sides and need an angle. If neither rule seems to fit immediately, consider whether you can calculate a missing angle first (angles in a triangle sum to 180 degrees).

Area of a triangle using sine

The formula Area = (1/2)ab sin(C) calculates the area of any triangle when you know two sides and the included angle. This formula is provided on the igcse 4mb1 trigonometry exam formula sheet, but knowing it from memory saves time.

Worked example: Find the area of a triangle with sides 8 cm and 11 cm and an included angle of 63 degrees.

Area = (1/2)(8)(11)sin(63) = 44 x 0.8910 = 39.20 cm² (to 4 s.f.).

Angles of elevation and depression

The angle of elevation is the angle measured upward from the horizontal to a line of sight. The angle of depression is measured downward from the horizontal. Both are always measured from the horizontal, not from the vertical.

Worked example: From the top of a cliff 45 m high, the angle of depression to a boat at sea is 32 degrees. How far is the boat from the base of the cliff?

The angle of depression from the cliff top equals the angle of elevation from the boat (alternate angles). In the right-angled triangle formed by the cliff, the horizontal distance, and the line of sight:

tan(32) = 45/d, where d is the horizontal distance.

d = 45/tan(32) = 45/0.6249 = 72.0 m (to 3 s.f.).

Bearings

Bearings are measured clockwise from north as three-digit numbers. Due east is 090, due south is 180, due west is 270. The edexcel igcse mathematics specification b revision notes for bearings should emphasise these conventions, as incorrect measurement is a frequent source of error.

Worked example: Town B is 20 km from town A on a bearing of 065 degrees. Town C is 30 km from town A on a bearing of 130 degrees. Find the distance BC and the bearing of C from B.

The angle BAC = 130 - 65 = 65 degrees.

Using the cosine rule: BC² = 20² + 30² - 2(20)(30)cos(65).

BC² = 400 + 900 - 1200 x 0.4226 = 1300 - 507.1 = 792.9.

BC = sqrt(792.9) = 28.16 km (to 4 s.f.).

To find the bearing of C from B, first find angle ABC using the sine rule:

sin(ABC)/30 = sin(65)/28.16.

sin(ABC) = 30 x sin(65)/28.16 = 30 x 0.9063/28.16 = 0.9655.

ABC = arcsin(0.9655) = 74.9 degrees.

The bearing of C from B = 065 + 180 - 74.9 = 170.1 degrees, which rounds to 170 degrees (to 3 s.f.).

Three-dimensional problems

The specification includes solving problems in three dimensions by calculation. These problems typically require you to identify a right-angled triangle within a 3D figure, apply Pythagoras or trigonometry to that triangle, and then use the result in a second calculation.

Worked example: A vertical flagpole stands at corner A of a horizontal rectangular field ABCD. AB = 20 m and BC = 15 m. The angle of elevation of the top of the flagpole from C is 18 degrees. Find the height of the flagpole.

AC is the diagonal of the rectangle: AC = sqrt(20² + 15²) = sqrt(625) = 25 m.

Let h be the height of the flagpole. In the right-angled triangle formed by A, C, and the top of the pole:

tan(18) = h/25.

h = 25 x tan(18) = 25 x 0.3249 = 8.12 m (to 3 s.f.).

Common mistakes in Trigonometry

Calculator in the wrong mode. The edexcel igcse mathematics specification b notes assume degrees, not radians. Check your calculator is set to degrees before every trigonometry question. A single wrong mode setting can invalidate an entire paper.
Using SOHCAHTOA in non-right-angled triangles. SOHCAHTOA only applies to right-angled triangles. For any other triangle, use the sine rule or cosine rule.
Ignoring the ambiguous case. When the sine rule gives sin(A) = k, there may be two valid angles. Always check whether the obtuse angle is geometrically possible.
Bearings measured anticlockwise or from the wrong direction. Bearings are always measured clockwise from north. A bearing of 045 means 45 degrees clockwise from north, not from east or west.
Rounding prematurely. In multi-step problems (especially bearings), keep full precision until the final answer. Rounding intermediate values causes cumulative error that shifts the final answer outside the acceptable range.

Self-check questions

Work through these edexcel igcse mathematics specification b practice questions fully before checking.

In a right-angled triangle, the hypotenuse is 20 cm and one of the acute angles is 35 degrees. Find the lengths of both shorter sides.
In triangle PQR, p = 14, q = 9, and angle R = 110 degrees. Find side r and the area of the triangle.
A ship sails 12 km on a bearing of 040, then 18 km on a bearing of 150. Find the direct distance from the starting point to the final position.
From a point 50 m from the base of a tower, the angle of elevation to the top is 28 degrees. Find the height of the tower.

Solutions

1. Side opposite 35 degrees = 20 x sin(35) = 20 x 0.5736 = 11.47 cm. Side adjacent to 35 degrees = 20 x cos(35) = 20 x 0.8192 = 16.38 cm.

2. r² = 196 + 81 - 2(14)(9)cos(110) = 277 - 252(-0.3420) = 277 + 86.18 = 363.18. r = sqrt(363.18) = 19.06 cm. Area = (1/2)(14)(9)sin(110) = 63 x 0.9397 = 59.2 cm² (to 3 s.f.).

3. Angle between the two legs = 150 - 40 = 110 degrees. Using the cosine rule: d² = 144 + 324 - 2(12)(18)cos(110) = 468 - 432(-0.3420) = 468 + 147.7 = 615.7. d = sqrt(615.7) = 24.8 km (to 3 s.f.).

4. tan(28) = h/50. h = 50 x tan(28) = 50 x 0.5317 = 26.6 m (to 3 s.f.).

Trigonometry in the edexcel igcse mathematics specification b explained framework rewards systematic, step-by-step working. Identify the triangle. Label the sides relative to the angle you are using. Select the appropriate rule. Execute the calculation. State the answer with correct units and appropriate rounding. Each of these steps is straightforward in isolation; the skill lies in chaining them together accurately under exam conditions.