Scottish examsHigher

Higher Maths formula sheet: what the SQA list gives you and what you must learn

Published 12 Nov 20268 min read

Higher Maths: Algebra & Functions

Higher Maths: Calculus

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Formula sheet

The card below opens today's Higher Maths paper with this list in the drawer, so you can practise with it beside you. The contents follow in our own layout. The list was the same in the 2025 and 2026 papers. Check it against the list printed in a recent paper, such as the 2026 Paper 1, before you rely on this page.

Formula sheet

Higher Maths: Algebra & Functions

Open formula sheet

Circle

x^2+y^2+2gx+2fy+c=0general equation of a circle

r=\sqrt{g^2+f^2-c}its centre is (−g, −f) and its radius is

(x-a)^2+(y-b)^2=r^2circle with centre (a, b) and radius r

Scalar product

a\cdot b=|a||b|\cos\thetascalar product from the angle θ between the vectors

a\cdot b=a_1b_1+a_2b_2+a_3b_3scalar product from components

Trigonometric formulae

\sin(A\pm B)=\sin A\cos B\pm\cos A\sin Bsine of a sum or difference

\cos(A\pm B)=\cos A\cos B\mp\sin A\sin Bcosine of a sum or difference

\sin2A=2\sin A\cos Adouble angle, sine

\cos2A=\cos^2A-\sin^2Adouble angle, cosine

\cos2A=2\cos^2A-1double angle, cosine, second form

\cos2A=1-2\sin^2Adouble angle, cosine, third form

Standard derivatives

\frac{d}{dx}(\sin ax)=a\cos axderivative of sin ax

\frac{d}{dx}(\cos ax)=-a\sin axderivative of cos ax

Standard integrals

\int\sin ax\ dx=-\frac{1}{a}\cos ax+cintegral of sin ax

\int\cos ax\ dx=\frac{1}{a}\sin ax+cintegral of cos ax

In the first circle equation, g and f are numbers, so 2g is the coefficient of x and 2f the coefficient of y. In the scalar product, θ is the angle between the two vectors when they are placed tail to tail.

What is not on the Higher Maths formula sheet

Most of Higher Maths is not on the list, so most of your revision is learning formulae and methods. These are all missing from it:

Straight lines. The gradient formula, y − b = m(x − a), the gradient as tan θ, and the rules for parallel and perpendicular lines.
Distance and midpoint. Both are needed for circles and for lines, and neither is printed.
Quadratics. The quadratic formula and the discriminant b² − 4ac are not given at Higher, though the National 5 list printed the quadratic formula.
Polynomials and logarithms. The factor and remainder theorems, the log laws, and how to change between exponential and log form.
Sequences. Recurrence relations and the limit of one.
Vectors. The length of a vector, the section formula, and the rules for unit vectors and collinear points.
Trigonometry. Exact values for 30°, 45° and 60°, radians, and the wave function. The list gives the addition formulae, but you decide when to use them.
Calculus. Differentiating and integrating powers of x, the chain rule, stationary points, and the area between curves. Only the sin ax and cos ax results are printed.

Try one of the formulae you must know. The quadratic formula is not on the Higher list, so check you can use it from memory in the Algebra course, then press Mark.

Press Mark to have your answer marked in the Algebra course.

How the papers use the list

Higher Mathematics is two question papers and no assignment, so the list is printed in both. The course specification gives 55 marks for Paper 1, where you may not use a calculator, and 65 for Paper 2, where you may. In the 2027 timetable, Paper 1 is on Friday 14 May from 09:00 to 10:15 and Paper 2 from 10:45 to 12:15 (Exam Timetable 2027, version 02).

Paper	Marks	Time	Calculator
Paper 1	55	1 hour 15 minutes	No
Paper 2	65	1 hour 30 minutes	Yes

Because Paper 1 has no calculator, a lot of the list is aimed at it: exact values, the scalar product with whole numbers, and double angle formulae with surds all turn up there. Our Higher Maths guide covers the course, the grades and the 2026 boundaries.

Using the formulae list under exam pressure

Read the question and decide the topic before you look at the list.
Find the line you need and copy it into your working. Writing it down earns the mark for the method.
Substitute, and keep exact values on Paper 1.
Check the form of the answer: an angle in degrees, a centre as a coordinate pair, an integral with "+ c".

Circle: centre and radius

Find the centre and radius of the circle with equation

x^2+y^2-6x+4y-12=0

Compare with x² + y² + 2gx + 2fy + c = 0. Then 2g = −6, so g = −3, and 2f = 4, so f = 2, with c = −12. The centre is (−g, −f) = (3, −2).

r=\sqrt{g^2+f^2-c}=\sqrt{9+4+12}=\sqrt{25}=5

Scalar product: the angle between two vectors

Find the angle between a = (1, 2, 2) and b = (2, 3, 6). From components, a·b = 1×2 + 2×3 + 2×6 = 20. The lengths are |a| = 3 and |b| = 7, so

\cos\theta=\frac{a\cdot b}{|a||b|}=\frac{20}{21}\quad\text{so}\quad\theta=17.8^\circ\ \text{(1 d.p.)}

Double angle formula

If sin x = 3/5 and x is acute, find sin 2x and cos 2x. The right triangle gives cos x = 4/5, so

\sin2x=2\sin x\cos x=2\times\frac{3}{5}\times\frac{4}{5}=\frac{24}{25}

\cos2x=1-2\sin^2x=1-2\times\frac{9}{25}=\frac{7}{25}

Choosing the form 1 − 2sin²x is the shortcut: it needs only sin x, which the question gave you.

Differentiating and integrating sin and cos

Differentiate 4 sin 2x. The list gives d/dx (sin ax) = a cos ax, so the answer is 4 × 2 cos 2x = 8 cos 2x. Check it in the Derivative Calculator, which gives 8 at x = 0.

Now evaluate the integral of cos 4x from 0 to π/12. The list gives ∫cos ax dx = (1/a) sin ax + c, so

\int_0^{\pi/12}\cos4x\ dx=\left[\frac{1}{4}\sin4x\right]_0^{\pi/12}=\frac{1}{4}\sin\frac{\pi}{3}-\frac{1}{4}\sin0=\frac{\sqrt{3}}{8}

How the Higher list compares with National 5 and Advanced Higher

Each level prints its own list, and the lists grow with the level. The National 5 list is mostly geometry and the quadratic formula, which Higher no longer prints. The Advanced Higher list is longer and adds series, the binomial theorem and complex numbers.

Level	What is printed
National 5	Quadratic formula, sine rule, cosine rule, area of a triangle, volumes of a sphere, cone and pyramid, standard deviation
Higher	Equations of a circle, scalar product, addition and double angle formulae, derivatives and integrals of sin ax and cos ax
Advanced Higher	Derivatives and integrals of inverse trig, tan, sec, cosec, cot, ln and e^x functions; sums of series; binomial theorem; Maclaurin series; de Moivre's theorem; vector product; rotation matrix

See the National 5 Maths formula sheet and the Advanced Higher Maths formula sheet for each in full.

Exam answers

The answers and a worked mark scheme come with today's paper once it is marked. Sit the paper with the list open beside you, mark it, and the worked answers show which formula each question wanted. We do not publish official or past paper answers.

Use the list with today's paper

Uni-Sine publishes free daily papers for Higher Maths: Algebra and Functions and Higher Maths: Calculus. Each is a short timed paper of 8 questions worth 25 to 35 marks, not a full SQA paper. Many schools sit a longer prelim in winter: see our prelims guide. Open the list, set a timer, and write the formula you use at the start of each answer.

Common mistakes with the list

Forgetting the factor a. Differentiating sin 5x gives 5 cos 5x, and integrating cos 4x brings in a factor of 1/4.
Losing the minus sign. The derivative of cos ax is negative, and so is the integral of sin ax.
Reading the centre as (g, f). The centre of the general circle is (−g, −f).
Taking c as the radius. In the general form, c is only a constant: the radius is √(g² + f² − c).
Using degrees for calculus. Integrals and derivatives of trig functions assume radians.
Dropping + c. An indefinite integral needs it.

Common questions

Do you get a formula sheet in SQA Higher Maths?

Yes. A formulae list is printed at the front of both Higher Mathematics papers. It is short, so most formulae still have to be learned.

What is on the Higher Maths formulae list?

The two equations of a circle, the scalar product, the addition and double angle formulae, and the standard derivatives and integrals of sin ax and cos ax.

Is the quadratic formula on the Higher Maths formula sheet?

No. It was on the National 5 list but is not printed at Higher, so you must know it.

Are the straight line and vector formulae given?

No. The gradient, the equation of a line, distance, midpoint, the length of a vector and the section formula are not on the list.

Does the list change from year to year?

It was the same in the 2025 and 2026 papers. Check the front of a recent paper for the current version.

Can I take my own formula sheet into the exam?

No. You use the list printed in the paper.

Revision cards

Say the answer first, then flip the card to check it.

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