Trigonometry and Pythagoras' theorem

Q: Do I need to memorize SOH-CAH-TOA and Pythagoras' theorem for the exam?

Yes! These are not provided on the formula sheet, so you must recall them. Higher-tier formulae (sine rule, cosine rule, area (1/2)ab sin C) are given in the exam booklet.

Q: What mode should my calculator be in for trig questions?

Always set your calculator to degrees mode for all Edexcel IGCSE Maths A trigonometry questions. Radian mode is not used in this specification.

Edexcel International GCSE Mathematics A· 4.8· 25 min read

1. Pythagoras' Theorem for 2D and 3D Triangles★★☆☆☆⏱ 5 min

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📘 Definition

Pythagoras' Theorem

a^2 + b^2 = c^2

For a right-angled triangle with hypotenuse (longest side, opposite right angle) $c$ , and shorter sides $a$ and $b$ , the sum of squares of the shorter sides equals the square of the hypotenuse.

Example:

Right triangle with sides 3cm, 4cm: hypotenuse = 5cm, since $3^{2} + 4^{2} = 5^{2}$ .

Foundation tier only covers 2D applications of Pythagoras' theorem, while Higher tier extends this to 3D shapes (cuboids, pyramids, prisms) by identifying right triangles inside the 3D shape.

📐 Worked Example

A cuboid has length 6cm, width 4cm, height 3cm. Find the length of the space diagonal of the cuboid (Higher only).

1
First find the diagonal of the base rectangle using 2D Pythagoras: base diagonal $d_{1}^{2} = 6^{2} + 4^{2} = 36 + 16 = 52$ .
2
Form a right triangle with $d_{1}$ , height 3cm, and space diagonal $d$ : $d^{2} = d_{1}^{2} + 3^{2} = 52 + 9 = 61$ .
3
Calculate $d = 61 \approx 7.81$ cm (3 significant figures).

Exam tip:

Always label the hypotenuse clearly to avoid mixing up sides when applying Pythagoras, and check your final value is longer than the other two sides.

2. Right-Angled Trigonometry (SOH-CAH-TOA)★★★☆☆⏱ 6 min

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📘 Definition

SOH-CAH-TOA

Mnemonic for the three trigonometric ratios for acute angles in right-angled triangles: SOH: $sin θ = \frac{Opposite}{Hypotenuse}$ ; CAH: $cos θ = \frac{Adjacent}{Hypotenuse}$ ; TOA: $tan θ = \frac{Opposite}{Adjacent}$ .

Foundation tier uses these ratios only for acute angles in 2D right triangles, while Higher tier extends use to obtuse angles (note: $sin (obtuse) = sin (180 - θ)$ , $cos (obtuse)$ is negative).

📐 Worked Example

A right triangle has hypotenuse 12cm, and an acute angle of 35°. Find the length of the side opposite the 35° angle.

1
Label sides: hypotenuse =12cm, opposite side = $x$ , angle $θ = 3 5^{\circ}$ .
2
We have hypotenuse and need opposite, so use sine (SOH): $sin 3 5^{\circ} = \frac{x}{12}$ .
3
Rearrange: $x = 12 \times sin 3 5^{\circ} \approx 12 \times 0.5736 \approx 6.88$ cm (3 sf).

Exam tip:

For Higher tier, if you calculate an obtuse angle using inverse sine, remember to check if 180° minus your result is the correct answer for the triangle.

3. 2D Trigonometry Problem Solving★★★☆☆⏱ 5 min

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Foundation tier problems include bearings (three-digit angles measured clockwise from north), while Higher tier adds angles of elevation (angle from horizontal up to an object) and depression (angle from horizontal down to an object, equal to elevation due to alternate angles).

📐 Worked Example

A ship sails 15km east from port A, then 8km north to port B. Find the bearing of B from A (Foundation tier).

1
Draw a right triangle with east side 15km, north side 8km, right angle at the turning point.
2
Find the angle between the north line and AB: $tan θ = \frac{15}{8} = 1.875$ .
3
Calculate $θ = arctan (1.875) \approx 61. 9^{\circ}$ , so bearing = 062° (three-digit format).

4. Sine Rule, Cosine Rule and Triangle Area Formula (Higher Only)★★★★☆Higher only⏱ 6 min

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For non-right angled triangles, use the sine rule, cosine rule, and area formula provided on your exam formula sheet. Use the sine rule when you have a matching side-angle pair, and the cosine rule when you have three sides or two sides and an included angle.

📐 Worked Example

A triangle has sides $a = 7$ cm, $b = 9$ cm, and included angle $C = 4 0^{\circ}$ . Find the area of the triangle, and the length of side $c$ .

1
Calculate area: $\frac{1}{2} ab sin C = 0.5 \times 7 \times 9 \times sin 4 0^{\circ} \approx 20.2$ cm² (3 sf).
2
Use cosine rule to find $c$ : $c^{2} = a^{2} + b^{2} - 2 ab cos C = 7^{2} + 9^{2} - 2 \times 7 \times 9 \times cos 4 0^{\circ}$ .
3
Simplify: $c^{2} = 49 + 81 - 126 \times 0.7660 \approx 33.5$ , so $c \approx 5.79$ cm (3 sf).

Exam tip:

The ambiguous case of the sine rule only occurs when you are finding an angle opposite a shorter side, so always check both possible acute and obtuse solutions.

5. 3D Trigonometry (Higher Only)★★★★☆Higher only⏱ 4 min

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Higher tier requires you to find the angle between a line and a plane. This is the smallest angle between the line and its projection onto the plane, forming a right triangle with the line, its projection, and the perpendicular height from the end of the line to the plane.

📐 Worked Example

A square-based pyramid has base side length 10cm, and vertical height 12cm. Find the angle between one of the slant edges and the base plane.

1
The projection of the slant edge onto the base is half the diagonal of the square base: half-diagonal = $52 \approx 7.071$ cm.
2
The right triangle has opposite side = height 12cm, adjacent = $52$ cm, angle $θ$ between slant edge and base.
3
$tan θ = \frac{12}{5 2} \approx 1.697$ , so $θ \approx 59. 5^{\circ}$ (1 dp).

Exam tip:

Always identify the right triangle explicitly in 3D problems, labelling all three sides clearly to avoid mixing up trig ratios.

6. Common Pitfalls

Wrong move:

Mixing up the hypotenuse with a shorter side when applying Pythagoras' theorem.

Why:

The hypotenuse is always the longest side, so it must be the $c$ term in $a^{2} + b^{2} = c^{2}$ .

Correct move:

Label the hypotenuse (opposite the right angle) first, then plug values into the formula, checking that $c$ is the largest value.

Wrong move:

Using calculator in radians mode for trig calculations.

Why:

All Edexcel IGCSE Maths A angles are in degrees, so radian mode will give incorrect values for sin, cos, tan.

Correct move:

Check your calculator shows "D" or "DEG" on the display before starting any trig question.

Wrong move:

Rounding intermediate calculation values too early.

Why:

Early rounding leads to inaccuracies in the final answer, which can cost marks.

Correct move:

Keep full calculator precision for all intermediate steps, only round the final answer to the required number of significant figures (usually 3 sf) or decimal places.

Wrong move:

Forgetting to write bearings as three-digit numbers.

Why:

Bearings are defined as three-digit angles measured clockwise from north, so missing leading zeros will lose marks.

Correct move:

Add leading zeros to bearings less than 100°, e.g. 62° becomes 062°, 8° becomes 008°.

Wrong move:

Only considering the acute angle solution when using the sine rule for obtuse angles.

Why:

$sin θ = sin (18 0^{\circ} - θ)$ , so there are two possible angles for any positive sine value between 0 and 1.

Correct move:

Check if the obtuse solution (180° minus the acute result) sums to less than 180° with the other known angles in the triangle, and use the appropriate value.

Wrong move:

Using non-included angles for the $\frac{1}{2} ab sin C$ area formula.

Why:

The formula only works when angle $C$ is the included angle between sides $a$ and $b$ .

Correct move:

Ensure the angle you use is between the two known sides, or rearrange the triangle to use an included angle if needed.

7. Quick Reference Cheatsheet

Concept	Formula/Rule	Tier	Recall/Given
Pythagoras' Theorem (2D)	$a^{2} + b^{2} = c^{2}$ , $c$ = hypotenuse	Foundation	Recall
SOH-CAH-TOA Ratios	$sin θ = O / H$ , $cos θ = A / H$ , $tan θ = O / A$	Foundation	Recall
Bearings	3-digit, clockwise from north	Foundation	Recall
Pythagoras' Theorem (3D cuboid)	$d = l^{2} + w^{2} + h^{2}$	Higher	Recall/Derive
Sine Rule	$\frac{a}{s i n A} = \frac{b}{s i n B} = \frac{c}{s i n C}$	Higher	Given
Cosine Rule	$a^{2} = b^{2} + c^{2} - 2 b c cos A$	Higher	Given
Triangle Area (non-right)	$\frac{1}{2} ab sin C$ (included angle $C$ )	Higher	Given
Angle of elevation/depression	Equal due to alternate angles	Higher	Recall

8. Frequently Asked

Do I need to memorize SOH-CAH-TOA and Pythagoras' theorem for the exam?

Yes! These are not provided on the formula sheet, so you must recall them. Higher-tier formulae (sine rule, cosine rule, area (1/2)ab sin C) are given in the exam booklet.

What mode should my calculator be in for trig questions?

Always set your calculator to degrees mode for all Edexcel IGCSE Maths A trigonometry questions. Radian mode is not used in this specification.

What is the ambiguous case for the sine rule?

When you use the sine rule to find an angle with sin θ = x where 0 < x < 1, there are two possible solutions: an acute angle (θ = arcsin x) and an obtuse angle (θ = 180° - arcsin x). Check if the obtuse angle is valid for the triangle sum (all angles add to 180°) to avoid mistakes.

Going deeper

syllabusEdexcel IGCSE Maths A 4MA1 SpecOfficial 2016 specification for reference

What's Next

Now that you have mastered trigonometry and Pythagoras' theorem for Edexcel IGCSE Maths A, you are ready to apply these skills to more complex geometry problems, including circle theorems and vector calculations, which often combine with trigonometry in higher-mark exam questions. Be sure to practice as many past paper questions as possible for both Foundation and Higher tiers, focusing on showing all your working clearly to maximise marks, and always checking your answers for reasonableness (e.g., the hypotenuse should always be the longest side of a right triangle). Pay special attention to ambiguous sine rule cases and 3D trigonometry problems, which are common discriminators for higher grades in the Higher tier paper.