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Modular
Fractions
Combinatorics
Primes
Pythagorean
Divisors & Euler
Trig Values
Trig Formulas
Bases

Modular Exponentiation

Compute large powers or modular inverses.

Fraction Calculator

Add, subtract, multiply or divide fractions.
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Combinations & Permutations

\(C(n,k)=\frac{n!}{k!(n-k)!}\) · \(A(n,k)=\frac{n!}{(n-k)!}\)

Prime Numbers

Generate all prime numbers up to a given limit.

Pythagorean Triples

Primitive triples \((a,b,c)\) with \(a^2+b^2=c^2\), sorted by \(c\).

Divisors & Euler's Totient

Prime factorization, divisor count, divisor sum, and \(\varphi(n)\).

Exact Trigonometric Values

Exact values every 15°, by quadrant. ↔ Scroll horizontally if needed.
Quadrant I · 0° → 90°
°0153045607590
\(\pi\)0\(\frac{\pi}{12}\)\(\frac{\pi}{6}\)\(\frac{\pi}{4}\)\(\frac{\pi}{3}\)\(\frac{5\pi}{12}\)\(\frac{\pi}{2}\)
\(\approx\)0.0000.2620.5240.7851.0471.3091.571
\(\sin\)0\(\frac{\sqrt{6}-\sqrt{2}}{4}\)\(\frac12\)\(\frac{\sqrt2}{2}\)\(\frac{\sqrt3}{2}\)\(\frac{\sqrt{6}+\sqrt{2}}{4}\)1
\(\cos\)1\(\frac{\sqrt{6}+\sqrt{2}}{4}\)\(\frac{\sqrt3}{2}\)\(\frac{\sqrt2}{2}\)\(\frac12\)\(\frac{\sqrt{6}-\sqrt{2}}{4}\)0
\(\tan\)0\(2-\sqrt3\)\(\frac{\sqrt3}{3}\)1\(\sqrt3\)\(2+\sqrt3\)∞
Quadrant II · 90° → 180°
°90105120135150165180
\(\pi\)\(\frac{\pi}{2}\)\(\frac{7\pi}{12}\)\(\frac{2\pi}{3}\)\(\frac{3\pi}{4}\)\(\frac{5\pi}{6}\)\(\frac{11\pi}{12}\)\(\pi\)
\(\approx\)1.5711.8332.0942.3562.6182.8803.142
\(\sin\)1\(\frac{\sqrt{6}+\sqrt{2}}{4}\)\(\frac{\sqrt3}{2}\)\(\frac{\sqrt2}{2}\)\(\frac12\)\(\frac{\sqrt{6}-\sqrt{2}}{4}\)0
\(\cos\)0\(-\frac{\sqrt{6}-\sqrt{2}}{4}\)\(-\frac12\)\(-\frac{\sqrt2}{2}\)\(-\frac{\sqrt3}{2}\)\(-\frac{\sqrt{6}+\sqrt{2}}{4}\)\(-1\)
\(\tan\)∞\(-(2+\sqrt3)\)\(-\sqrt3\)\(-1\)\(-\frac{\sqrt3}{3}\)\(-(2-\sqrt3)\)0
Quadrant III · 180° → 270°
°180195210225240255270
\(\pi\)\(\pi\)\(\frac{13\pi}{12}\)\(\frac{7\pi}{6}\)\(\frac{5\pi}{4}\)\(\frac{4\pi}{3}\)\(\frac{17\pi}{12}\)\(\frac{3\pi}{2}\)
\(\approx\)3.1423.4033.6653.9274.1894.4514.712
\(\sin\)0\(-\frac{\sqrt{6}-\sqrt{2}}{4}\)\(-\frac12\)\(-\frac{\sqrt2}{2}\)\(-\frac{\sqrt3}{2}\)\(-\frac{\sqrt{6}+\sqrt{2}}{4}\)\(-1\)
\(\cos\)\(-1\)\(-\frac{\sqrt{6}+\sqrt{2}}{4}\)\(-\frac{\sqrt3}{2}\)\(-\frac{\sqrt2}{2}\)\(-\frac12\)\(-\frac{\sqrt{6}-\sqrt{2}}{4}\)0
\(\tan\)0\(2-\sqrt3\)\(\frac{\sqrt3}{3}\)1\(\sqrt3\)\(2+\sqrt3\)∞
Quadrant IV · 270° → 360°
°270285300315330345360
\(\pi\)\(\frac{3\pi}{2}\)\(\frac{19\pi}{12}\)\(\frac{5\pi}{3}\)\(\frac{7\pi}{4}\)\(\frac{11\pi}{6}\)\(\frac{23\pi}{12}\)\(2\pi\)
\(\approx\)4.7124.9745.2365.4985.7606.0216.283
\(\sin\)\(-1\)\(-\frac{\sqrt{6}+\sqrt{2}}{4}\)\(-\frac{\sqrt3}{2}\)\(-\frac{\sqrt2}{2}\)\(-\frac12\)\(-\frac{\sqrt{6}-\sqrt{2}}{4}\)0
\(\cos\)0\(\frac{\sqrt{6}-\sqrt{2}}{4}\)\(\frac12\)\(\frac{\sqrt2}{2}\)\(\frac{\sqrt3}{2}\)\(\frac{\sqrt{6}+\sqrt{2}}{4}\)1
\(\tan\)∞\(-(2+\sqrt3)\)\(-\sqrt3\)\(-1\)\(-\frac{\sqrt3}{3}\)\(-(2-\sqrt3)\)0

Trigonometric Formulas

⚛ Fundamental Identities
① \(\sin^2\alpha + \cos^2\alpha = 1\)
② \(1 + \tan^2\alpha = \frac{1}{\cos^2\alpha}\)
③ \(1 + \cot^2\alpha = \frac{1}{\sin^2\alpha}\)
④ \(\tan\alpha \cdot \cot\alpha = 1\)
⑤ \(\tan\alpha = \frac{\sin\alpha}{\cos\alpha} \quad,\quad \cot\alpha = \frac{\cos\alpha}{\sin\alpha}\)
📊 Signs by Quadrant
QuadAnglesincostancot
I0° – 90°++++
II90° – 180°+−−−
III180° – 270°−−++
IV270° – 360°−+−−
🔄 Reduction to Quadrant I
Key Rule: If the argument is \(\frac{\pi}{2} \pm \alpha\) or \(\frac{3\pi}{2} \pm \alpha\) → the function changes (\(\sin \leftrightarrow \cos\), \(\tan \leftrightarrow \cot\)).
If it is \(\pi \pm \alpha\) or \(2\pi \pm \alpha\) → the function remains the same; sign depends on the quadrant.
Expressionsincostancot
\(\frac{\pi}{2} - \alpha\)\(\cos \alpha\)\(\sin \alpha\)\(\cot \alpha\)\(\tan \alpha\)
\(\frac{\pi}{2} + \alpha\)\(\cos \alpha\)\(-\sin \alpha\)\(-\cot \alpha\)\(-\tan \alpha\)
\(\pi - \alpha\)\(\sin \alpha\)\(-\cos \alpha\)\(-\tan \alpha\)\(-\cot \alpha\)
\(\pi + \alpha\)\(-\sin \alpha\)\(-\cos \alpha\)\(\tan \alpha\)\(\cot \alpha\)
\(\frac{3\pi}{2} - \alpha\)\(-\cos \alpha\)\(-\sin \alpha\)\(\cot \alpha\)\(\tan \alpha\)
\(\frac{3\pi}{2} + \alpha\)\(-\cos \alpha\)\(\sin \alpha\)\(-\cot \alpha\)\(-\tan \alpha\)
\(2\pi - \alpha\)\(-\sin \alpha\)\(\cos \alpha\)\(-\tan \alpha\)\(-\cot \alpha\)
\(2\pi + \alpha\)\(\sin \alpha\)\(\cos \alpha\)\(\tan \alpha\)\(\cot \alpha\)
📐 Basic Formulas
➕ Sum (\(\alpha + \beta\))
\(\sin(\alpha+\beta)=\) \(\sin\alpha\cos\beta + \cos\alpha\sin\beta\)
\(\cos(\alpha+\beta)=\) \(\cos\alpha\cos\beta - \sin\alpha\sin\beta\)
\(\tan(\alpha+\beta)=\) \(\frac{\tan\alpha + \tan\beta}{1 - \tan\alpha\tan\beta}\)
\(\cot(\alpha+\beta)=\) \(\frac{\cot\alpha\cot\beta - 1}{\cot\beta + \cot\alpha}\)
➖ Difference (\(\alpha - \beta\))
\(\sin(\alpha-\beta)=\) \(\sin\alpha\cos\beta - \cos\alpha\sin\beta\)
\(\cos(\alpha-\beta)=\) \(\cos\alpha\cos\beta + \sin\alpha\sin\beta\)
\(\tan(\alpha-\beta)=\) \(\frac{\tan\alpha - \tan\beta}{1 + \tan\alpha\tan\beta}\)
\(\cot(\alpha-\beta)=\) \(\frac{\cot\alpha\cot\beta + 1}{\cot\beta - \cot\alpha}\)
×2 Double Angle (\(2\alpha\))
\(\sin 2\alpha =\) \(2\sin\alpha\cos\alpha\)
\(\cos 2\alpha =\) \(\cos^2\alpha - \sin^2\alpha = 2\cos^2\alpha - 1 = 1 - 2\sin^2\alpha\)
\(\tan 2\alpha =\) \(\frac{2\tan\alpha}{1 - \tan^2\alpha}\)
\(\cot 2\alpha =\) \(\frac{\cot^2\alpha - 1}{2\cot\alpha}\)
×3 Triple Angle (\(3\alpha\))
\(\sin 3\alpha =\) \(3\sin\alpha - 4\sin^3\alpha\)
\(\cos 3\alpha =\) \(4\cos^3\alpha - 3\cos\alpha\)
\(\tan 3\alpha =\) \(\frac{3\tan\alpha - \tan^3\alpha}{1 - 3\tan^2\alpha}\)
\(\cot 3\alpha =\) \(\frac{\cot^3\alpha - 3\cot\alpha}{3\cot^2\alpha - 1}\)
÷2 Half Angle (\(\alpha/2\))
\(\sin(\alpha/2) =\) \(\pm\sqrt{\frac{1 - \cos\alpha}{2}}\)
\(\cos(\alpha/2) =\) \(\pm\sqrt{\frac{1 + \cos\alpha}{2}}\)
\(\tan(\alpha/2) =\) \(\pm\sqrt{\frac{1-\cos\alpha}{1+\cos\alpha}} = \frac{\sin\alpha}{1+\cos\alpha} = \frac{1-\cos\alpha}{\sin\alpha}\)
\(\cot(\alpha/2) =\) \(\pm\sqrt{\frac{1+\cos\alpha}{1-\cos\alpha}} = \frac{\sin\alpha}{1-\cos\alpha} = \frac{1+\cos\alpha}{\sin\alpha}\)

Base Conversion

Convert numbers between bases 2–16. Supports digits 0-9 and letters A-F.