In triangle $XYZ$, $XY = 14$ cm, $YZ = 9$ cm and $\angle X = 30^\circ$. Find the two possible values of $\angle Z$ to 1 decimal place.

M1 for using the sine rule: $\frac{\sin Z}{14} = \frac{\sin 30^\circ}{9}$. M1 for calculating $\sin Z = \frac{14 \times 0.5}{9} = \frac{7}{9}$. A1 for the principal acute angle $Z = 51.1^\circ$. A1 for the alternative obtuse angle $Z = 180^\circ - 51.1^\circ = 128.9^\circ$.

A Level Math Revision

Edexcel IAL P1 Trigonometry: exam questions and answers, notes and worked examples

Updated October 2026 · By Pete Bromfield

P1 trigonometry covers the sine and cosine rules, the area of a triangle $\frac{1}{2} a b \sin C$ , radians with arc length and sector area, and the graphs of sin, cos and tan. Pick the rule from what you are given: two sides and the angle between them means the cosine rule; a side and its opposite angle means the sine rule. Arc and sector formulas need the angle in radians.

P1 · Pure Mathematics 1
Calculator allowed
3 worked examples
67 practice questions
Tested in 4 of the last 4 papers

Practise P1 trigonometry →

Notes
Formulas
Worked examples
Mistakes
FAQ

P1 Trigonometry notes

The sine rule (and the ambiguous case)

$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$ . Use it when you know a side and the angle opposite it. Finding an angle can give two answers, because $\sin θ = \sin (180^{\circ} - θ)$ : if $\sin B = 0.8$ then $B = {53.1}^{\circ}$ or $B = {126.9}^{\circ}$ . Keep the obtuse value only if the three angles still add up to less than $180^{\circ}$ with it.

The cosine rule

$a^{2} = b^{2} + c^{2} - 2 b c \cos A$ . Use it with two sides and the included angle, or with all three sides to find an angle: $\cos A = \frac{b^{2} + c^{2} - a^{2}}{2 b c}$ . In a triangle with sides 5, 7 and 8, the angle opposite the side of length 7 has $\cos A = \frac{25 + 64 - 49}{80} = \frac{1}{2}$ , so it is $60^{\circ}$ .

Area of a triangle

Area $= \frac{1}{2} a b \sin C$ , where $C$ is the angle between sides $a$ and $b$ . With $a = 6$ , $b = 10$ and $C = 30^{\circ}$ the area is $15$ .

Radians, arcs and sectors

$π$ radians $= 180^{\circ}$ , so 1 radian is about ${57.3}^{\circ}$ . With $θ$ in radians, arc length $s = r θ$ and sector area $A = \frac{1}{2} r^{2} θ$ . For $r = 6$ and $θ = 1.2$ : arc $7.2$ and sector area $21.6$ . A segment is the sector minus the triangle: $\frac{1}{2} r^{2} (θ - \sin θ)$ .

Trig graphs, symmetry and exact values

$y = \sin x$ and $y = \cos x$ repeat every $360^{\circ}$ ( $2 π$ ); $y = \tan x$ repeats every $180^{\circ}$ and has asymptotes at $x = 90^{\circ}$ and $x = 270^{\circ}$ . Use the symmetries $\sin (180^{\circ} - θ) = \sin θ$ and $\cos (360^{\circ} - θ) = \cos θ$ to find second solutions. Know the exact values $\sin 30^{\circ} = \frac{1}{2}$ , $\cos 30^{\circ} = \frac{\sqrt{3}}{2}$ , $\tan 45^{\circ} = 1$ and $\sin 60^{\circ} = \frac{\sqrt{3}}{2}$ .

Key formulas

Sine rule Learn it: $\frac{a}{\sin A} = \frac{b}{\sin B}$
Cosine rule In the formula booklet: $a^{2} = b^{2} + c^{2} - 2 b c \cos A$
Area; arc; sector (radians) Learn it: $\frac{1}{2} a b \sin C; s = r θ; \frac{1}{2} r^{2} θ$

In the formula booklet = printed in Pearson's IAL formulae booklet, so know where to find it. Learn it = not printed there, so learn it by heart. All P1 formulas on one page (free PDF) →

P1 Trigonometry worked examples

Each mark scheme is checked line by line before it is published here. M marks are for method, A marks for accurate answers that follow from them, B marks for independent results.

Worked example 1 · 5 marks · hard

Triangle $A B C$ has sides $A B = 4$ cm and $B C = 5$ cm, with angle $A B C = 60^{\circ}$ . Triangle $A C D$ shares side $A C$ and has a right angle at $C$ . If $C D = 7$ cm, find the exact length of $A D$ .

Mark scheme

M1 for using the cosine rule in triangle ABC to find $A C^{2}$ : $A C^{2} = 4^{2} + 5^{2} - 2 (4) (5) \cos 60^{\circ}$ .
M1 for evaluating to get $A C^{2} = 16 + 25 - 40 (0.5) = 21$ .
M1 for recognising that triangle $A C D$ is right-angled at $C$ , so Pythagoras' theorem applies: $A D^{2} = A C^{2} + C D^{2}$ .
M1 for substituting the known values: $A D^{2} = 21 + 7^{2} = 21 + 49 = 70$ .
A1 for finding $A D = \sqrt{70}$ cm.

5 steps · 1 to fill in

Worked example 2 · 5 marks · hard

Two identical circles of radius $8$ cm intersect such that the circumference of each passes through the centre of the other. Find the exact area of the overlapping region.

Mark scheme

M1 for recognising that lines drawn from the intersection points to the centres form two equilateral triangles, hence the central angle in each circle covering the overlap is $\frac{2 π}{3}$ .
M1 for establishing the overlapping region consists of two identical segments, so Area $= 2 \times \frac{1}{2} (8^{2}) (\frac{2 π}{3} - \sin \frac{2 π}{3})$ .
M1 for substituting $\sin \frac{2 π}{3} = \frac{\sqrt{3}}{2}$ .
M1 for expanding the expression $64 (\frac{2 π}{3} - \frac{\sqrt{3}}{2})$ .
A1 for the exact simplified area $\frac{128 π}{3} - 32 \sqrt{3} {cm}^{2}$ .

5 steps

Worked example 3 · 4 marks · hard

In triangle $X Y Z$ , $X Y = 14$ cm, $Y Z = 9$ cm and $∠ X = 30^{\circ}$ . Find the two possible values of $∠ Z$ to 1 decimal place.

Mark scheme

M1 for using the sine rule: $\frac{\sin Z}{14} = \frac{\sin 30^{\circ}}{9}$ .
M1 for calculating $\sin Z = \frac{14 \times 0.5}{9} = \frac{7}{9}$ .
A1 for the principal acute angle $Z = {51.1}^{\circ}$ .
A1 for the alternative obtuse angle $Z = 180^{\circ} - {51.1}^{\circ} = {128.9}^{\circ}$ .

4 steps · 1 to fill in

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Common P1 trigonometry mistakes

Calculator in the wrong mode. Degrees for triangle questions given in degrees, radians for arcs and sectors. A wrong mode gives a confident wrong answer, so glance at the screen before every trig calculation.
Missing the second angle. When the sine rule gives an angle, check whether $180^{\circ}$ minus it also fits the triangle. Questions that say "two possible values" expect both.
Arc length with the angle in degrees. $s = r θ$ and $\frac{1}{2} r^{2} θ$ only work with $θ$ in radians. Convert first: multiply degrees by $\frac{π}{180}$ .
Rushing the cosine rule. Work out $b^{2} + c^{2} - 2 b c \cos A$ as one value and then square root it. Taking $(b^{2} + c^{2} - 2 b c)$ first and multiplying by $\cos A$ is a common slip.

Every P1 mistake from the examiner reports, with the paper it came from →

On your calculator: how to draw trig graphs and read off max, min and solutions. A calculator is allowed in every IAL unit.

P1 Trigonometry FAQ

Is the sine rule in the IAL formula booklet?

On our IAL formula sheet the cosine rule is marked In the formula booklet, while the sine rule and the area and sector formulas are marked Learn it. Learn those by heart.

Do I use radians or degrees in P1?

Both appear. Triangle questions are usually in degrees, but the arc length and sector area formulas s = rθ and ½r²θ need the angle in radians.

How can I tell if my calculator is in radians?

Check the angle setting shown on the screen, or test it: sin 0.5 is about 0.479 in radians but about 0.00873 in degrees. With the angle in radians, a sector of radius 6 cm and angle 0.7 has arc 4.2 cm and area 12.6 cm².

Is a calculator allowed in IAL P1?

Yes. A calculator is allowed in every Edexcel IAL maths unit, P1 included. Method marks are for the working you write down, so show it.

How long is the IAL P1 exam?

Each IAL maths unit, P1 included, is one written paper of 1 hour 30 minutes worth 75 marks.

How often does trigonometry come up in P1?

In our mapping of the 4 Pearson P1 papers from June 2024 to June 2025, trigonometry was tested in 4 of them (6 questions in all). Any topic can be examined in any series.

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More P1: Algebra · Coordinate Geometry · Differentiation · Integration · Graphs and Transformations

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Other units: IAL P2 · IAL P3 · IAL P4 · IAL S1 · IAL M1

Edexcel IAL P1 Trigonometry: exam questions and answers, notes and worked examples

Try it first

P1 Trigonometry notes

The sine rule (and the ambiguous case)

The cosine rule

Area of a triangle

Radians, arcs and sectors

Trig graphs, symmetry and exact values

Key formulas

P1 Trigonometry worked examples

Worked example 1 · 5 marks · hard

Mark scheme

Worked example 2 · 5 marks · hard

Mark scheme

Worked example 3 · 4 marks · hard

Mark scheme

Common P1 trigonometry mistakes

P1 Trigonometry FAQ

Is the sine rule in the IAL formula booklet?

Do I use radians or degrees in P1?

How can I tell if my calculator is in radians?

Is a calculator allowed in IAL P1?

How long is the IAL P1 exam?

How often does trigonometry come up in P1?

Next steps

Your P1 revision kit