Reading time: ~10 minutes | Suitable for: Students Grades 9–12, All Curricula (IB, IGCSE, GCSE, A-Level, AP, CBSE)

Whether you’re halfway through a homework set, building a revision sheet before an exam, or just trying to remember which identity actually applies to the problem in front of you — this guide has every major trigonometry formula and identity in one organised place, grouped the way they’re actually used, not just listed alphabetically.

Trigonometry shows up across nearly every curriculum students study internationally — IB Mathematics, IGCSE and GCSE Maths, A-Level Pure Mathematics, AP Precalculus, and CBSE/ICSE — and while the formulas themselves don’t change between curricula, how they’re tested often does. We’ve noted where that matters.

Basic Trigonometric Ratios
Reciprocal, Quotient & Pythagorean Identities
Co-function and Even-Odd Identities
Sum and Difference Formulas
Double Angle and Half Angle Formulas
Product-to-Sum and Sum-to-Product Formulas
Law of Sines, Cosines, and Tangents
The Unit Circle: Standard Angle Values
How These Formulas Show Up by Curriculum
Common Mistakes Students Make
How to Actually Memorize These
FAQs

1. Basic Trigonometric Ratios

These come from a right-angled triangle and are the foundation everything else builds on.

Ratio	Formula	Memory Aid
sin θ	Opposite / Hypotenuse	SOH
cos θ	Adjacent / Hypotenuse	CAH
tan θ	Opposite / Adjacent	TOA

Expert Insight: Students who memorise SOH-CAH-TOA without ever sketching the triangle it describes often apply it incorrectly the moment a question doesn’t draw the triangle for them. Drawing it yourself first — every time — is the habit that prevents most basic trig errors.

2. Reciprocal, Quotient, and Pythagorean Identities

These three groups are the backbone of trig identity work, especially for proof-style and simplification questions.

Reciprocal Identities

Identity
csc θ = 1 / sin θ
sec θ = 1 / cos θ
cot θ = 1 / tan θ

Quotient Identities

Identity
tan θ = sin θ / cos θ
cot θ = cos θ / sin θ

Pythagorean Identities

Identity
sin²θ + cos²θ = 1
1 + tan²θ = sec²θ
1 + cot²θ = csc²θ

The first Pythagorean identity is the one to know cold — the other two are simply derived by dividing through by cos²θ or sin²θ. Understanding that derivation (rather than memorising all three as separate facts) makes them far easier to recall under exam pressure.

3. Co-function and Even-Odd Identities

Co-function Identities

Describe the relationship between complementary angles (angles that sum to 90°):

Identity
sin(90° − θ) = cos θ
cos(90° − θ) = sin θ
tan(90° − θ) = cot θ
csc(90° − θ) = sec θ
sec(90° − θ) = csc θ
cot(90° − θ) = tan θ

Even-Odd Identities

Identity	Type
sin(−θ) = −sin θ	Odd
cos(−θ) = cos θ	Even
tan(−θ) = −tan θ	Odd

4. Sum and Difference Formulas

These let you find the sine, cosine, or tangent of an angle made by adding or subtracting two known angles.

Formula
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) / (1 ∓ tan A tan B)

Case Study: A-Level Pure Maths Student

A student consistently lost marks on questions asking for exact values like sin(75°), not realising 75° could be expressed as 45° + 30° and solved using the sum formula with known values. Once the connection clicked — “an unfamiliar angle is often just two familiar angles combined” — exact-value questions stopped being a guessing game.

5. Double Angle and Half Angle Formulas

Double angle formulas come directly from the sum formulas above, with A = B.

Double Angle Formulas

Formula
sin 2θ = 2 sin θ cos θ
cos 2θ = cos²θ − sin²θ = 2cos²θ − 1 = 1 − 2sin²θ
tan 2θ = 2 tan θ / (1 − tan²θ)

Half Angle Formulas

Formula
sin(θ/2) = ±√[(1 − cos θ) / 2]
cos(θ/2) = ±√[(1 + cos θ) / 2]
tan(θ/2) = (1 − cos θ) / sin θ = sin θ / (1 + cos θ)

The three versions of cos 2θ aren’t three separate facts to memorise — they’re the same identity rewritten using sin²θ + cos²θ = 1. Picking the right version for a given problem (usually whichever one eliminates the variable you don’t have) is the actual skill being tested.

6. Product-to-Sum and Sum-to-Product Formulas

Less frequently tested at IGCSE/GCSE level, but common in IB HL, A-Level, and AP contexts involving calculus or wave/signal applications.

Product-to-Sum

Formula
sin A cos B = ½[sin(A+B) + sin(A−B)]
cos A sin B = ½[sin(A+B) − sin(A−B)]
cos A cos B = ½[cos(A−B) + cos(A+B)]
sin A sin B = ½[cos(A−B) − cos(A+B)]

Sum-to-Product

Formula
sin A + sin B = 2 sin((A+B)/2) cos((A−B)/2)
sin A − sin B = 2 cos((A+B)/2) sin((A−B)/2)
cos A + cos B = 2 cos((A+B)/2) cos((A−B)/2)
cos A − cos B = −2 sin((A+B)/2) sin((A−B)/2)

7. Law of Sines, Law of Cosines, and Law of Tangents

These apply to any triangle — not just right-angled ones — which is what makes them essential beyond basic trig.

Law	Formula	Best used when
Law of Sines	a/sin A = b/sin B = c/sin C	You know an angle and its opposite side, plus one more piece of info
Law of Cosines	c² = a² + b² − 2ab cos C	You know two sides and the included angle, or all three sides
Law of Tangents	(a−b)/(a+b) = tan[(A−B)/2] / tan[(A+B)/2]	Less commonly required; useful in some A-Level/IB HL triangle problems

8. The Unit Circle: Standard Angle Values

These values come up so frequently that memorising them directly saves significant time in exams.

θ (degrees)	θ (radians)	sin θ	cos θ	tan θ
0°	0	0	1	0
30°	π/6	1/2	√3/2	1/√3
45°	π/4	√2/2	√2/2	1
60°	π/3	√3/2	1/2	√3
90°	π/2	1	0	undefined

9. How These Formulas Show Up by Curriculum

Expert Insight: A meaningful number of students who “know all the formulas” still underperform in trigonometry — not because they’re missing a formula, but because each curriculum’s exam style asks them to select and combine formulas differently.

Curriculum	Typical emphasis
IGCSE / GCSE Maths	Basic ratios, Pythagorean identity, Law of Sines/Cosines in applied geometry problems
IB Mathematics (AA/AI)	Heavier use of identities for proof, plus radians and unit circle work integrated with calculus
A-Level Pure Maths	Sum/difference, double angle, and half angle formulas used in proof and equation-solving questions
AP Precalculus / Calc	Identities used as tools to simplify expressions before differentiation or integration
CBSE / ICSE	Strong emphasis on identity proofs and exact-value problems using standard angles

If your child is transitioning between curricula — for instance, moving from a CBSE-style proof-heavy approach into IB’s more application-integrated style — the formulas themselves transfer directly; what usually needs rebuilding is comfort with the new question format, not the content.

10. Common Mistakes Students Make

Mixing up co-function and reciprocal identities — cos(90° − θ) = sin θ is not the same relationship as sec θ = 1/cos θ, but they’re commonly confused under time pressure.
Forgetting the ± in half-angle formulas — the sign depends on which quadrant θ/2 falls in, and dropping it is one of the most common silent errors in exam answers.
Treating “identity” and “equation” the same way — an identity is true for all θ; an equation needs to be solved for specific values. Applying identity logic to an equation question (or vice versa) is a frequent source of lost marks.
Not simplifying before substituting — many “impossible-looking” exam questions become solvable in one line once a Pythagorean or quotient identity is applied first.

11. How to Actually Memorise These (Not Just Stare at Them)

Flashcards work for some students, but trigonometry rewards derivation-based memory far more than rote memory — because if you forget one formula, understanding where it comes from lets you rebuild it on the spot.

Learn the sum formulas properly, and the double angle formulas become free (just set A = B).
Learn the first Pythagorean identity, and the other two follow by dividing through.
Practice deriving, not just reciting — covering the formula and rebuilding it from the unit circle or triangle, repeatedly, builds far more durable recall than flashcards alone.

Case Study: Year 10 IGCSE Student

A student who consistently froze on identity-proof questions made noticeable progress not from memorising more formulas, but from practising the habit of writing down every related identity at the start of a question before attempting to manipulate anything — turning a blank page into a toolbox before starting the actual proof.

FAQs

Q: What is the difference between a trigonometric formula and a trigonometric identity?

An identity is an equation that's true for every value of the angle (such as sin²θ + cos²θ = 1), while a formula is typically a tool for calculating a specific unknown, such as the Law of Cosines for finding a triangle's missing side.

Q: What are the three basic trigonometric ratios?

The three basic ratios are sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent), commonly remembered through the mnemonic SOH-CAH-TOA.

Q: How do you derive the double angle formulas?

Double angle formulas come directly from the sum formulas by setting both angles equal — for example, sin 2θ = sin(θ + θ) = sin θ cos θ + cos θ sin θ = 2 sin θ cos θ.

Q: Do I need to memorise the unit circle, or can I derive it?

Most students benefit from memorising the values for 0°, 30°, 45°, 60°, and 90°, since deriving them from scratch in an exam setting costs valuable time; understanding the underlying triangles, however, makes the memorisation far more durable.

Q: Are trigonometry formulas the same across IB, IGCSE, A-Level, and AP?

Yes — the mathematical formulas themselves are universal, but each curriculum emphasises and tests them differently, which is why exam technique (not just formula recall) often determines performance.

Q: Why do I keep getting trig identity proofs wrong even though I know the formulas?

This usually points to a gap in strategy rather than knowledge — proofs typically require starting from the more complex side of an equation and simplifying toward the simpler side using the identities above, a specific technique that benefits from guided practice.

Q: What's the easiest way to remember the Pythagorean identities?

Learn sin²θ + cos²θ = 1 thoroughly, then derive the other two by dividing the entire equation by cos²θ (to get 1 + tan²θ = sec²θ) or sin²θ (to get 1 + cot²θ = csc²θ), rather than memorising all three independently.

Q: When should I use the Law of Sines versus the Law of Cosines?

Use the Law of Sines when you know an angle and its opposite side; use the Law of Cosines when you know two sides and the angle between them, or all three sides and need to find an angle.

Conclusion

Trigonometry formulas aren’t really a list to memorise in isolation — they’re a connected system, where a handful of core identities (the Pythagorean identity, the sum formulas, and the basic ratios) generate almost everything else through simple derivation. Once a student sees that structure instead of forty unconnected facts, trigonometry stops being a memory test and starts being what it actually is: a logical, learnable system.

If your child has the formulas in front of them but still struggles to know which one to use and when — that’s a strategy and practice gap, not a memory gap, and it’s exactly the kind of thing a specialist trigonometry tutor can close quickly with the right guided practice.

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Author: Academic Director, The Gurukul Global | Reviewed by: Mathematics & IB Specialist | Last Updated: June 2026 | Sources: IB, Cambridge International, and College Board curriculum specifications for Mathematics

List of Trigonometry Formulas and Identities (Complete Guide)

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