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Trigonometry: A Level Maths knowledge organiser

Everything to know about trigonometry on one page: key definitions, the formulas, a worked example, the mistakes to avoid and a checklist of what you should be able to do.

Download the A4 sheet (PDF)

Key definitions

Radian
The angle that cuts off an arc equal to the radius; π radians = 180°.
Identity
An equation true for every value of the variable, such as sin²θ + cos²θ = 1.
Principal value
The single value your calculator gives for an inverse trig function; find the others from the graph.
Harmonic form
a sin θ + b cos θ written as R sin(θ + α), to find maximum and minimum values.

Key formulas

Formulas marked with a label are given in the exam (we only say so where our formula sheet confirms it). Learn the rest.

Sine rule; cosine rule\(\frac a{\sin A}=\frac b{\sin B},\) \(a^2=b^2+c^2-2bc\cos A\)
Area; arc; sector (radians)\(\tfrac12ab\sin C;\) \(s=r\theta;\) \(A=\tfrac12r^2\theta\)
Identities\(\tan\theta=\frac{\sin\theta}{\cos\theta},\) \(\sin^2\theta+\cos^2\theta=1\)
Reciprocal\(\sec^2\theta=1+\tan^2\theta,\) \(\cosec^2\theta=1+\cot^2\theta\)
Double angle\(\sin2A=2\sin A\cos A,\) \(\cos2A=\cos^2A-\sin^2A=2\cos^2A-1=1-2\sin^2A\)
Double angle (tan)\(\tan2A=\frac{2\tan A}{1-\tan^2A}\)
Compound sinIn the formula booklet: Edexcel, AQA, OCR\(\sin(A\pm B)=\sin A\cos B\) \(\pm\cos A\sin B\)
Compound cosIn the formula booklet: Edexcel, AQA, OCR\(\cos(A\pm B)=\cos A\cos B\) \(\mp\sin A\sin B\)
Compound tanIn the formula booklet: Edexcel, AQA, OCR\(\tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\tan B}\)
Harmonic form, \(R>0\)\(a\sin\theta+b\cos\theta=R\sin(\theta+\alpha),\) \(R=\sqrt{a^2+b^2},\) \(\tan\alpha=\tfrac ba\)
Small angles (radians)In the formula booklet: Edexcel, AQA, OCR\(\sin\theta\approx\theta,\) \(\cos\theta\approx1-\tfrac{\theta^2}2,\) \(\tan\theta\approx\theta\)
Exact values\(\sin30^\circ=\tfrac12,\) \(\sin45^\circ=\tfrac{\sqrt2}2,\) \(\sin60^\circ=\tfrac{\sqrt3}2,\) \(\tan30^\circ=\tfrac1{\sqrt3},\) \(\tan60^\circ=\sqrt3\)
Inverse trig ranges\(\arcsin\in[-\tfrac\pi2,\tfrac\pi2],\) \(\arccos\in[0,\pi],\) \(\arctan\in(-\tfrac\pi2,\tfrac\pi2)\)
Radians\(\pi\text{ rad}=180^\circ\)

Worked example

Solve 2sin²x = sin x for 0° ≤ x ≤ 360°.

  1. 2sin²x − sin x = 0, so sin x(2 sin x − 1) = 0
  2. sin x = 0 gives x = 0°, 180°, 360°
  3. sin x = ½ gives x = 30°, 150°

Answer: x = 0°, 30°, 150°, 180°, 360°

Common mistakes

  • R cos(θ − α): wrong ratio for tan α and calculator in the wrong mode
  • Small-angle approximations: missing brackets
  • (2x)³ written as 2x³
  • Wrong nCr index

More on what examiners see students get wrong: Examiner Insights.

You should be able to…

  • Use the cosine rule a² = b² + c² − 2bc cos A to find a missing side or a missing angle in a triangle that is not right-angled.
  • Sketch transformations of trigonometric graphs such as y = 2 sin x, y = cos(x + 30°) and y = tan 2x, and describe how the period and amplitude change.
  • Solve equations that are quadratic in sin or cos, such as 2sin²x − sin x − 1 = 0, using identities and factorising to find every solution.
  • Define secant, cosecant and cotangent as reciprocals of cosine, sine and tangent, and find their exact values for standard angles.
  • Apply the angle addition formulae to prove identities and to solve equations such as sin(x + 30°) = 2 cos x in a given interval.
  • Model periodic situations such as tides and Ferris wheels with trigonometric functions, finding greatest and least values and the times they occur.

The printable sheet

Trigonometry knowledge organiser for A Level Maths: one A4 page of key definitions, formulas, a worked example and common mistakes
Trigonometry knowledge organiser (A Level Maths), A4. Download the PDF.

Revise it next

Other A Level Maths topics: Proof · Algebra and functions · Coordinate geometry · Sequences and series · Exponentials and logarithms · Differentiation · Integration · Numerical methods · Vectors · Sampling and data · Probability · Binomial and normal distributions · Hypothesis testing · Kinematics · Forces and Newton's laws · Moments · All A Level Maths organisers