Trigonometry formulas link the sin, cos and tan of angles. Learn three core formulas well, and you can rebuild most of the others.
Learn three. Build the rest.
Core 1
sin²θ + cos²θ = 1
Core 2
sin(A + B) = sin A cos B + cos A sin B
Core 3
cos(A + B) = cos A cos B − sin A sin B
What are trigonometry formulas?
Trigonometry formulas are rules that hold for every angle. They swap one trig expression for an easier one.
Core 1 is the Pythagoras theorem on a right-angled triangle with a hypotenuse of 1. Exact values are on our unit circle page (coming soon).
Derive the trigonometry formulas from 3 core ones
Divide core 1 by cos²θ: 1 + tan²θ = sec²θ. By sin²θ: 1 + cot²θ = cosec²θ.
Put −B for B in cores 2 and 3, using sin(−B) = −sin B and cos(−B) = cos B. This gives sin(A − B) and cos(A − B).
Put B = A: sin 2A = 2 sin A cos A and cos 2A = cos²A − sin²A. Core 1 turns this into 2cos²A − 1 or 1 − 2sin²A.
Divide core 2 by core 3, then top and bottom by cos A cos B, for tan(A + B).
Add sin(A + B) and sin(A − B) for product to sum. Read it backwards for sum to product.
Not sure of a sign?
Test with A = B. cos(A − B) = cos 0 = 1 needs cos²A + sin²A, so the sign is plus.
How to use trigonometry formulas: 3 worked uses
Use 1: simplify (1 − cos 2x) ÷ sin 2x
1 − cos 2x = 2sin²x, and sin 2x = 2 sin x cos x.
2sin²x ÷ (2 sin x cos x) = tan x.
Use 2: prove (sin x + cos x)² = 1 + sin 2x
Expand the left side: sin²x + 2 sin x cos x + cos²x.
Core 1 makes this 1 + 2 sin x cos x, which is 1 + sin 2x.
Use 3: solve cos 2x = cos x for 0° ≤ x < 360°
Use cos 2x = 2cos²x − 1: 2cos²x − cos x − 1 = 0, so (2cos x + 1)(cos x − 1) = 0.
cos x = 1 gives 0°. cos x = −1/2 gives 120° and 240°. So x = 0°, 120° or 240°.
The trigonometry formulas sheet, grouped by use
1. Ratios
tan θ = sin θ/cos θsec θ = 1/cos θcosec θ = 1/sin θcot θ = 1/tan θ
You want all in sin and cos.
2. Squares
sin²θ + cos²θ = 11 + tan²θ = sec²θ1 + cot²θ = cosec²θ
You know one ratio and need another.
3. Minus and 90° − θ
sin(−θ) = −sin θcos(−θ) = cos θsin(90° − θ) = cos θcos(90° − θ) = sin θ
Two angles add to 90°.
4. Compound angle
sin(A ± B) = sin A cos B ± cos A sin Bcos(A ± B) = cos A cos B ∓ sin A sin Btan(A ± B) = (tan A ± tan B)/(1 ∓ tan A tan B)
Angles like 75°: sin(45° + 30°) = (√6 + √2)/4.
5. Double angle
sin 2A = 2 sin A cos Acos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²Atan 2A = 2tan A/(1 − tan²A)
One line mixes 2x and x.
6. Power down
sin²A = (1 − cos 2A)/2cos²A = (1 + cos 2A)/2
You must remove a square, as in calculus.
7. Product to sum
2 sin A cos B = sin(A + B) + sin(A − B)2 cos A cos B = cos(A + B) + cos(A − B)2 sin A sin B = cos(A − B) − cos(A + B)
Two trig terms are multiplied.
8. Sum to product
sin C + sin D = 2 sin((C + D)/2) cos((C − D)/2)sin C − sin D = 2 cos((C + D)/2) sin((C − D)/2)cos C + cos D = 2 cos((C + D)/2) cos((C − D)/2)cos C − cos D = −2 sin((C + D)/2) sin((C − D)/2)
A sum of sines or cosines equals 0.
Our advice: SAT, groups 1 to 3. JEE Main, all eight.
Common trigonometry formula mistakes
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- sin(A + B) = sin A + sin B. But sin 60° ≈ 0.866, and sin 30° + sin 30° = 1: CNC, concept not clear.
- A plus in cos(A + B). It has a minus: CNC, concept not clear.
- Copies tan θ as cos θ/sin θ, which is cot θ: CWV, copied a wrong value.
- Drops 2 sin x cos x from (sin x + cos x)²: CMT, calculation mistake.
- Skips steps in a proof. Name each formula you use: BPT, bad presentation.
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Test prep corner
Trigonometry formulas in SAT and JEE Main maths
The digital SAT reference sheet gives no trigonometry identities. For trigonometry, it shows only the two special right triangles. For JEE Main, Trigonometry is one of 14 units in NTA’s maths topic list. These are SAT-style and JEE-style practice questions, not real past questions.
SAT-style: θ is an acute angle in a right triangle, and cos θ = 5/13. What is tan θ?
sin²θ = 1 − 25/169 = 144/169, so sin θ = 12/13. tan θ = sin θ ÷ cos θ = 12/5. Trap: 5/12 is cot θ.
JEE-style: What is the greatest value of 3 sin x + 4 cos x?
Take α with cos α = 3/5 and sin α = 4/5. By core 2, the sum is 5 sin(x + α). Sin is at most 1, so the answer is 5. Trap: 3 + 4 = 7.
Practise more with our SAT-style maths practice questions. See the JEE Main maths marking scheme.
Facts checked on 11 November 2026 on satsuite.collegeboard.org and jeemain.nta.nic.in.
Quick check: 5 trigonometry formula questions
1. Which is equal to sin 2A?
Not quite. Test it at 30°.
Yes! Put B = A in core 2.
Not quite. That is −cos 2A.
2. Which is equal to cos(A − B)?
Not quite. That is cos(A + B).
Not quite. Cos does not split.
Yes! At A = B it gives 1.
3. 1 + tan²θ is equal to:
Yes! Divide core 1 by cos²θ.
Not quite. cosec²θ is 1 + cot²θ.
Not quite. Divide core 1 by cos²θ.
4. cos²x − sin²x simplifies to:
Not quite. 1 needs a plus.
Yes! The first form of cos 2x.
Not quite. sin 2x is 2 sin x cos x.
5. sin 3x + sin x is equal to:
Not quite. Sin does not add like that.
Not quite. (3x + x)/2 = 2x.
Yes! Sum to product, C = 3x, D = x.
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Questions people ask about trigonometry formulas
What are the basic trigonometry formulas?
sin²θ + cos²θ = 1, tan θ = sin θ/cos θ, sin(A + B) and cos(A + B).
How do I remember them?
Learn the three core formulas. Rebuild the rest, then test each with A = B.
Which formulas does the SAT give?
None of the identities. The sheet shows the special right triangles.
Which formulas are most used in JEE Main?
NTA does not publish this. Groups 2, 4 and 5 sit under most steps.
Is there a printable sheet?
Yes. Print the formula sheet section of this page.
Keep exploring
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