Advanced trigonometry for A level — identities, equations and the sine and cosine rules

At GCSE you learnt trigonometry in a right-angled triangle — the sine, cosine and tangent of an acute angle, plus the sine and cosine rules. At A level the bar rises significantly: now we talk about the unit circle and angles of any size, about radians, about trigonometric identities that have to be proved and applied, about solving trigonometric equations with infinitely many solutions, and about using the sine and cosine rules in modelling. This guide goes through all these tools step by step, with a table of special values, full examples at exam level, a list of the common mistakes and a practice plan that will take you to full command.

A-level trigonometry is a real step up from what you learnt at GCSE. There, everything was inside a right-angled triangle: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, and that was that. At A level, by contrast, sine and cosine become full functions defined for every angle — angles bigger than 90°, negative angles too — through the unit circle. On top of that come three new and weighty tools: trigonometric identities that let you simplify expressions and prove equalities, trigonometric equations that have infinitely many solutions and need to be written as a general formula, and the sine and cosine rules used in modelling problems that free us from the restriction of the right-angled triangle. This guide takes these topics one by one, explains the idea, gives the exact formula and demonstrates it in an exam-level question. At the end you will find the common mistakes and an organised practice plan.

The unit circle and special values

The unit circle is a circle centred at the origin (0, 0) with radius 1. Every angle θ is measured from the positive direction of the x-axis, anticlockwise. The point where the rotating arm meets the circle is (cos θ, sin θ) — that is, the horizontal coordinate of the point is the cosine, and the vertical coordinate is the sine. This is the definition that generalises trigonometry to every angle: now cos 120° or sin 210° are simply the coordinates of points on the circle.

From the circle you immediately get the signs of the functions in the quadrants. In the first quadrant (0° to 90°) both sine and cosine are positive. In the second quadrant (90° to 180°) sine is positive and cosine negative. In the third quadrant (180° to 270°) both are negative. In the fourth quadrant (270° to 360°) sine is negative and cosine positive. The memory aid is the CAST diagram: going anticlockwise from the fourth quadrant — Cos, All, Sin, Tan — names which function is positive in each quadrant.

The circle also yields the reduction identities (related angles). For example: sin(180° − θ) = sin θ, cos(180° − θ) = −cos θ, sin(−θ) = −sin θ, cos(−θ) = cos θ, sin(360° − θ) = −sin θ, cos(360° − θ) = cos θ. These identities let you bring any angle back to a calculation with a familiar acute angle.

Radians — the A-level measure of angle

Up to GCSE you measured angles in degrees. At A level a new measure comes in — the radian — and it is the measure you work in later for the calculus of trigonometric functions. One radian is the angle at the centre of the unit circle that subtends an arc of length 1 (that is, the length of the radius). Since the circumference of the unit circle is 2π, a full turn of 360° equals 2π radians.

The conversion rule: 180° = π radians. From this: to convert from degrees to radians multiply by π/180, and to convert from radians to degrees multiply by 180/π. The common values: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. It is worth knowing these by heart. Arc length is rθ and sector area is ½r²θ — with θ in radians.

The fundamental identity: sin²θ + cos²θ = 1

The most important identity in all of trigonometry is sin²θ + cos²θ = 1, hence the name 'the Pythagorean trigonometric identity'. It follows directly from Pythagoras' theorem on the unit circle: the point (cos θ, sin θ) lies on a circle of radius 1, so its distance from the origin is 1, that is cos²θ + sin²θ = 1². The identity holds for every angle θ without exception.

Two useful variations follow from the identity: sin²θ = 1 − cos²θ, and cos²θ = 1 − sin²θ. They let you, given one of the two, find the other (up to a sign decided by the quadrant). Example: if cos θ = 3/5 and θ is in the fourth quadrant, then sin²θ = 1 − 9/25 = 16/25, that is sin θ = ±4/5. In the fourth quadrant sine is negative, so sin θ = −4/5.

Another useful identity derived from it is tan θ = sin θ / cos θ (defined only when cos θ ≠ 0). If you divide the fundamental identity by cos²θ you get 1 + tan²θ = sec²θ — an identity that appears in proofs in the second year. These two identities, together with the fundamental identity, are the basis for every simplification of a trigonometric expression.

Compound-angle and double-angle identities

These are the identities that let you calculate a function of a sum or difference of angles, and expand expressions. The three compound-angle identities: sin(A ± B) = sin A · cos B ± cos A · sin B; cos(A + B) = cos A · cos B − sin A · sin B; cos(A − B) = cos A · cos B + sin A · sin B. Note the reversal of the sign in the cosine identity — a common mistake.

From the compound-angle identities you get the double-angle identities by substituting B = A: sin(2A) = 2 · sin A · cos A; cos(2A) = cos²A − sin²A. The double-angle cosine has two further forms obtained with the fundamental identity: cos(2A) = 2cos²A − 1 and also cos(2A) = 1 − 2sin²A. The three forms are equivalent, and you choose according to what is convenient in the question.

An example of calculating an exact value with a difference identity: cos 15° = cos(45° − 30°) = cos 45° · cos 30° + sin 45° · sin 30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4. That is how you find exact values of angles that are not in the basic table.

An example of using a double angle to simplify: the expression sin(2A)/(1 + cos(2A)). Substitute the identities: the numerator is 2 sin A cos A, and the denominator — using cos(2A) = 2cos²A − 1 — is 1 + 2cos²A − 1 = 2cos²A. The expression becomes (2 sin A cos A)/(2cos²A) = sin A / cos A = tan A. An elegant simplification that demonstrates the power of the identities.

Trigonometric equations — solutions in an interval and general solutions

A trigonometric equation is an equation in which the unknown sits inside a trigonometric function, for example sin x = 1/2. The difficulty is that such an equation has infinitely many solutions — because the functions are periodic. So there are two kinds of answer: a solution in a given interval (for example 0° ≤ x < 360°), where you return a finite list of solutions, and a general solution, where you write a formula that describes all the infinitely many solutions.

The general solution of sin x = a (where −1 ≤ a ≤ 1): x = α + 360°·k or x = 180° − α + 360°·k, where α is the principal solution (an angle for which sin α = a) and k is any integer. In radians: x = α + 2πk or x = π − α + 2πk.

The general solution of cos x = a: x = ±α + 360°·k (that is x = α + 360°k and also x = −α + 360°k), where cos α = a. In radians: x = ±α + 2πk. The general solution of tan x = a is simpler because tangent has a period of only 180°: x = α + 180°·k, and in radians x = α + πk.

Example — a solution in an interval. Solve sin x = 1/2 for 0° ≤ x < 360°. The principal solution is α = 30° (because sin 30° = 1/2). Sine is positive in the first and second quadrants, so the solutions in the interval are x = 30° and x = 180° − 30° = 150°. Two answers only.

Example — a general solution. Solve 2cos x = √3. Isolate: cos x = √3/2, so α = 30°. The general solution: x = ±30° + 360°·k, k ∈ ℤ. If you are also asked for the solutions between 0° and 360°, substitute k = 0 and k = 1 and get x = 30° and x = 330°.

Example — a quadratic in sine. Solve 2sin²x − sin x − 1 = 0. Let t = sin x and get 2t² − t − 1 = 0, whose solutions are t = 1 and t = −1/2. Going back: sin x = 1 gives x = 90° + 360°k; sin x = −1/2 gives x = −30° + 360°k or x = 210° + 360°k. A combination of algebra and trigonometry — a very common exam pattern.

The sine rule and the cosine rule — any triangle

So far all the trigonometry in a triangle required a right angle. The sine rule and the cosine rule free you from that restriction and let you solve any triangle. The convention: in a triangle you label the angles with the capital letters A, B, C and the sides opposite them with the small letters a, b, c respectively (a opposite A and so on).

The sine rule: a/sin A = b/sin B = c/sin C. That is, the ratio of each side to the sine of the angle opposite it is constant. When do you use it? When two angles and a side are given, or two sides and the angle opposite one of them. Example: in a triangle A = 40°, B = 60°, a = 8. Find b: b = a · sin B / sin A = 8 · sin 60° / sin 40° ≈ 8 · 0.866 / 0.643 ≈ 10.78.

The cosine rule: c² = a² + b² − 2ab · cos C (and similarly for each side). This is a generalisation of Pythagoras' theorem — when C = 90° we have cos C = 0 and the formula reduces to c² = a² + b². When do you use it? When two sides and the angle between them are given (then you calculate the third side), or when all three sides are given (then you calculate an angle).

An example of the cosine rule: in a triangle a = 5, b = 7 and the angle C between them is 60°. Calculate c: c² = 25 + 49 − 2·5·7·cos 60° = 74 − 70·(1/2) = 74 − 35 = 39, so c = √39 ≈ 6.24. The reverse example: if all three sides are given, rearrange the formula to cos C = (a² + b² − c²)/(2ab) and get the angle.

The area of a triangle: ½·a·b·sin C

When two sides and the angle between them are given, you do not need a height to calculate the area. The formula: area = ½ · a · b · sin C, where a and b are the two sides and C is the angle enclosed between them. The formula follows from the classic formula (½ · base · height): the height of the triangle equals a · sin C, and the base is b.

Example: a triangle in which a = 6, b = 10 and the angle between them C = 30°. The area: ½ · 6 · 10 · sin 30° = 30 · (1/2) = 15 square units. Note that the angle must be the one between the two given sides — if you take a different angle, the answer will be wrong.

Common mistakes

• Mixing degrees and radians on the calculator — check DEG/RAD at the start of every question.
• Forgetting the second solution of a sine or cosine equation — sine also has a solution at 180° − α, cosine also at −α.
• Forgetting the +360°k (or +2πk) in the general solution — without it the answer is incomplete.
• A wrong sign in the compound-angle cosine identity: cos(A + B) = cos A · cos B − sin A · sin B (minus!), unlike sine.
• A sign error in the quadrant — decide the sign of sine or cosine by the quadrant the angle is in.
• Choosing an angle that is not enclosed in the area formula ½ab·sin C — the angle must be between the two sides.
• Cancelling a trigonometric factor from both sides of an equation (for example dividing by cos x) and losing solutions — better to move everything to one side and factorise.

A practice plan

Week 1 — the unit circle, radians and special values. Learn the table of values, practise converting degrees ⇄ radians, and calculate the sine and cosine of angles in all four quadrants using the reduction identities.

Week 2 — identities. Twenty exercises on simplifying and proving identities, using the fundamental identity, tan = sin/cos and the double-angle identities repeatedly. The aim: to recognise at a glance which identity to apply.

Week 3 — equations. Practise solutions in an interval and general solutions, including equations that reduce to a quadratic (substituting t = sin x) and equations with a double angle. Always make sure you write the +360°k.

Week 4 — the general triangle. Ten triangle exercises: identify when the sine rule and when the cosine rule apply, calculate sides, angles and area. Finish with three full past-paper questions under exam conditions.

Summary

A-level trigonometry rests on four pillars: the unit circle (which generalises the functions to every angle), the identities (which let you simplify and prove), the equations (with the general solution) and the sine and cosine rules for area and for solving any triangle. Each of them appears in the Pure papers, and they are often combined in a single question. Command comes from systematic practice: learn the table of values and the identities, solve dozens of equations until the general solution becomes a reflex, and practise quickly recognising the right rule in any triangle. Do that, and trigonometry turns from the intimidating topic into the safest source of marks in the exam.

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