Trigonometry is one of the topics that stays with you from GCSE right through A level, and it combines remembering identities with solving equations and triangle calculations. In this guide we organise the whole topic: exact values at 30°, 45° and 60°, the unit circle, the fundamental identities, solving trigonometric equations with general solutions, and the sine rule and cosine rule for any triangle — including the common mistakes and a practice plan that will take you to full command.
Trigonometry is one of the most persistent topics in maths exams — it appears in the geometry of the triangle, in solving equations, and in applied questions. Many people fear it because it demands combining several things: remembering values and identities, using the unit circle, and solving equations. But once you organise the parts — exact values, identities, solving equations, and the sine and cosine rules — you discover that the topic is logical and consistent. In this guide we build everything from the base, with full examples for every step, so that you walk into the exam with confidence. Exact values and the sine and cosine rules are GCSE Higher content; the unit circle, identities and general solutions are A level.
Exact values — the angles you have to remember
At the base of everything are the exact values of sine, cosine and tangent at the common angles. These values must be known by heart — GCSE Higher examines them explicitly, and they are not on any formula sheet. The three important angles are 30°, 45° and 60°, joined by the extremes 0° and 90°. Note that these values are exact — you write √3/2, not 0.866. Writing a decimal approximation instead of the exact value can cost marks.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
The unit circle — beyond acute angles
In a right-angled triangle the angles are between 0° and 90°, but at A level you solve equations with larger angles too — up to 360° and beyond. This is where the unit circle comes in: a circle of radius 1, on which every angle corresponds to a point (cos, sin). The cosine is the x-coordinate of the point, and the sine is the y-coordinate. The circle is divided into four quadrants, and the signs of sine and cosine change between them.
| Quadrant | Angle range | sin | cos |
|---|---|---|---|
| First | 0°-90° | positive | positive |
| Second | 90°-180° | positive | negative |
| Third | 180°-270° | negative | negative |
| Fourth | 270°-360° | negative | positive |
A handy rule of thumb for the signs (the CAST diagram): in the first quadrant everything is positive; in the second only the sine is positive; in the third only the tangent is positive; in the fourth only the cosine is positive. The circle also explains useful identities such as sin(180° − x) = sin(x) and cos(360° − x) = cos(x), which help simplify expressions and solve equations.
The fundamental identities
The most important identity is the Pythagorean trigonometric identity: sin²x + cos²x = 1, true for every angle x. Two useful versions follow from it: sin²x = 1 − cos²x and cos²x = 1 − sin²x. This identity is the key to converting between sine and cosine inside an equation — for example when you want to express everything in terms of one variable.
Another important identity is tan x = sin x / cos x. It links the three ratios and lets you solve equations involving tangent by converting to sine and cosine. An example of using the Pythagorean identity: if sin x = 3/5 and x is in the first quadrant, then cos²x = 1 − 9/25 = 16/25, so cos x = 4/5 (positive because the quadrant is the first), and tan x = (3/5)/(4/5) = 3/4.
Solving trigonometric equations
A simple trigonometric equation has the form sin x = a or cos x = a. To solve it, first find a principal angle (one solution in the range 0°-90°), then use the unit circle to find all the solutions in the required range (usually 0°-360°). Example: solve sin x = 1/2 in the range 0°-360°. The principal angle is 30°. Sine is positive in the first and second quadrants, so the solutions are x = 30° and x = 180° − 30° = 150°.
When a general solution is required (all the solutions, not just those in one range), you add multiples of 360°. For sin x = 1/2: x = 30° + 360°k or x = 150° + 360°k, where k is an integer. For cos x = a the general solution is x = ±(principal angle) + 360°k, because cosine is symmetric about the x-axis.
More complex equations require using an identity before solving. Example: solve 2cos²x + cos x − 1 = 0. This is a quadratic in the variable cos x. Let t = cos x and we get 2t² + t − 1 = 0, giving (2t − 1)(t + 1) = 0, that is t = 1/2 or t = −1. Back-substituting: cos x = 1/2 gives x = 60° or x = 300°; cos x = −1 gives x = 180°. So one equation resolves into three solutions in the range 0°-360°.
The sine rule and the cosine rule
In a triangle that is not right-angled you cannot use the ordinary side ratios, so two rules are used. The sine rule: a/sin A = b/sin B = c/sin C, where a, b, c are the sides and A, B, C the angles opposite them. The rule is useful when you are given an angle and the side opposite it, plus one more piece of information. Example: in a triangle A = 40°, B = 60°, and a = 10. Find b: b = a·sin B / sin A = 10·sin 60° / sin 40° ≈ 10·0.866/0.643 ≈ 13.47.
The cosine rule: c² = a² + b² − 2ab·cos C. This is a generalisation of Pythagoras' theorem to any triangle (when C = 90° the cosine is zero and you are back to Pythagoras). You use it in two cases: when two sides and the angle between them are given (and you want the third side), or when all three sides are given (and you want an angle). Example: in a triangle a = 5, b = 7, and the angle between them C = 60°. Then c² = 25 + 49 − 2·5·7·cos 60° = 74 − 70·0.5 = 74 − 35 = 39, so c = √39 ≈ 6.24.
| When to use it | The rule | Formula |
|---|---|---|
| An angle and its opposite side, plus one more fact | The sine rule | a/sin A = b/sin B = c/sin C |
| Two sides and the angle between them | The cosine rule | c² = a² + b² − 2ab·cos C |
| Three sides, looking for an angle | The cosine rule | cos C = (a² + b² − c²)/(2ab) |
Common mistakes that lose marks
Mistake 1 — writing a decimal approximation instead of an exact value. When an exact value is required, sin 60° is √3/2 and not 0.87. Mistake 2 — missing solutions in a trigonometric equation. sin x = 1/2 gives two solutions in the range 0°-360° (30° and 150°), not one. Always check every quadrant where the sign fits.
Mistake 3 — forgetting the general solution. When the question asks for all the solutions, you must add +360°k. Mistake 4 — choosing the wrong rule in a triangle: the sine rule needs a pair of an angle and its opposite side; if two sides and the angle between them are given — that is the cosine rule. Mistake 5 — a wrong sign in the cosine rule: the formula is minus 2ab·cos C, and a sign slip wrecks the whole calculation.
A trigonometry practice plan
Trigonometry is a topic that combines memory and skill, so organised practice is essential. The following plan assumes about an hour a day and is built for about a fortnight.
- Days 1-2: exact values and the unit circle. Learn the table and practise identifying the signs in every quadrant.
- Days 3-4: the identities sin² + cos² = 1 and tan = sin/cos. Twenty simplification and conversion exercises.
- Days 5-6: simple trigonometric equations, including general solutions. Twenty exercises.
- Days 7-8: equations that need an identity (quadratics in sin or cos). Fifteen exercises.
- Days 9-10: the sine rule and the cosine rule in triangle problems and applied problems, under timed conditions.
Summary
Trigonometry only looks complicated until you organise it into parts: exact values that you remember by heart, the unit circle that explains the signs, the identity sin² + cos² = 1 that links sine and cosine, solving equations with a general solution, and the sine and cosine rules for any triangle. If you remember to write exact values, find all the solutions in every quadrant, and choose the right rule according to the information — trigonometry will become one of your strongest topics in the exam. The way there is through repeated practice, and that is exactly what is waiting for you on the MathsUK practice pages.
Frequently asked questions
Which trigonometric values do you need to know by heart?
You need to know sin, cos and tan of the angles 0°, 30°, 45°, 60° and 90°. For example sin 30° = 1/2, cos 30° = √3/2, sin 45° = cos 45° = √2/2, sin 60° = √3/2, cos 60° = 1/2. It is important to write the exact values (like √3/2) and not a decimal approximation (0.87), because an approximation can lose marks when an exact value is required. These are examined on GCSE Higher Paper 1 (non-calculator).
What is the most important trigonometric identity?
The identity sin²x + cos²x = 1, true for every angle. From it follow sin²x = 1 − cos²x and cos²x = 1 − sin²x, and it is the key to converting between sine and cosine inside an equation. Another important identity is tan x = sin x / cos x, which lets you solve equations with tangent by converting to sine and cosine.
How many solutions does an equation like sin x = 1/2 have?
In the range 0°-360° there are two solutions, because sine is positive in the first and second quadrants: x = 30° and x = 150°. If a general solution is required (all the solutions), you add multiples of 360°: x = 30° + 360°k and x = 150° + 360°k where k is an integer. Missing one of the solutions or the general solution is a common mistake.
When do you use the sine rule and when the cosine rule?
The sine rule (a/sin A = b/sin B = c/sin C) when a pair of an angle and its opposite side is given plus one more fact. The cosine rule (c² = a² + b² − 2ab·cos C) when two sides and the angle between them are given and you want the third side, or when all three sides are given and you want an angle. The cosine rule is a generalisation of Pythagoras' theorem to any triangle. Both rules are on the GCSE Higher specification.
What is the most common mistake in trigonometry questions?
Missing solutions in an equation (finding only one solution instead of all of them in every quadrant), forgetting the general solution +360°k when it is required, and writing a decimal approximation instead of an exact value. In triangle problems — choosing the wrong rule or a wrong sign in the cosine rule (minus 2ab·cos C). Repeated practice reduces these mistakes.
A graded set of exercises — from exact values to the sine rule and the cosine rule in exam style
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