Free Online Scientific Calculator — Trig, Log, Factorial & Memory

Expression Syntax Reference

This calculator accepts natural mathematical notation — the same style used in most programming languages — rather than requiring you to commit inputs one keystroke at a time like a pocket calculator. You can type an entire expression, review the live preview, and correct mistakes before hitting equals.

Parentheses group sub-expressions exactly as in arithmetic: (3 + 4) * 5 evaluates to 35, not 23. Nested parentheses work too: sqrt((3^2) + (4^2)) gives 5.

DEG vs. RAD Mode

The angle mode toggle — visible at the top right of the calculator — controls how trig functions interpret their argument. In DEG mode, sin(90) means sin of 90 degrees and returns 1. In RAD mode, sin(90) means sin of 90 radians, which is approximately 0.894. This is the most common source of unexpected results in trigonometry.

When to use DEG: geometry, navigation, engineering drawings, physics problems involving angles measured from a reference direction, and any textbook problem that states angles in degrees.

When to use RAD: calculus (derivatives and integrals of trig functions are cleanest in radians), signal processing, rotation matrices in computer graphics, and physics formulas involving angular frequency (ω = 2πf).

Converting between the two: multiply degrees by π/180 to get radians, or multiply radians by 180/π to get degrees. A full circle is 360° = 2π radians.

Trigonometric Functions and Their Inverses

The calculator provides all six primary trig operations split across two rows:

• sin, cos, tan — the standard forward functions: given an angle, return a ratio
• sin⁻¹, cos⁻¹, tan⁻¹ — the inverse functions: given a ratio, return the angle

Inverse trig functions are sometimes written arcsin, arccos, arctan in textbooks; they're the same thing. In DEG mode, sin⁻¹(0.5) returns 30 (degrees). In RAD mode it returns π/6 ≈ 0.5236.

Domain limits matter: asin and acos only accept inputs in [−1, 1] — outside this range you'll get NaN because no real angle has a sine greater than 1. atan accepts any real number and returns an angle in (−90°, 90°) in DEG mode or (−π/2, π/2) in RAD mode.

Common trig identities you can verify:

sin(x)^2 + cos(x)^2       → always 1 (Pythagorean identity)
sin(2*30) = 2*sin(30)*cos(30)  → 0.866 = 0.866 (double-angle)
tan(45) = sin(45)/cos(45)  → 1 = 1

Logarithms: ln vs. log₁₀

The calculator offers both logarithm bases because the two are used in fundamentally different domains:

ln (natural logarithm, base e) — used in continuous growth and decay models, probability distributions, calculus, and information theory. ln(1) = 0, ln(e) = 1, ln(e^3) = 3. The natural log appears in compound interest formulas, radioactive decay rates, and entropy calculations.

log₁₀ (common logarithm, base 10) — used in engineering, acoustics (decibels), chemistry (pH), and earthquake magnitude (Richter scale). log10(1) = 0, log10(10) = 1, log10(1000) = 3. Each unit on a log₁₀ scale represents a factor of 10.

Converting between bases: log base b of x = ln(x) / ln(b). So log base 2 of 1024 = log(1024) / log(2) = 10. Enter this as log(1024) / log(2) in the calculator.

Example — pH calculation: pH = −log₁₀([H⁺]). For a hydrogen ion concentration of 0.001 mol/L: 0 - log10(0.001) = 3. Battery acid at [H⁺] = 1 mol/L gives pH 0; pure water at [H⁺] = 10⁻⁷ gives pH 7.

Factorial and Combinatorics

The n! button appends a ! to the current expression. Factorial is defined for non-negative integers: 0! = 1, 1! = 1, 2! = 2, 3! = 6, and so on up to about 170! before exceeding floating-point range.

Combinations and permutations are built from factorials:

Combinations (choosing k items from n without order):

C(n, k) = n! / (k! × (n−k)!)
C(10, 3) = 10! / (3! × 7!) = 3628800 / (6 × 5040) = 120

Enter as: 10! / (3! * 7!)

Permutations (choosing k items from n with order):

P(n, k) = n! / (n−k)!
P(10, 3) = 10! / 7! = 720

Enter as: 10! / 7!

Stirling's approximation check: For large n, ln(n!) ≈ n*log(n) - n. For n=20: log(20!) - (20*log(20) - 20) shows the approximation error directly.

Memory Registers

The four memory buttons save and retrieve values across separate calculations without retyping intermediate results:

A blue dot on the MR button indicates memory holds a nonzero value.

Example — multi-step physics problem (kinetic energy then potential energy comparison):

1. Calculate kinetic energy: 0.5 * 2.5 * 14^2 = 245 joules → press M+ to store it
2. Calculate potential energy: 2.5 * 9.81 * 11.5 = 281.8 joules → press M+ to add it
3. Press MR to recall 526.8 — the total mechanical energy — and continue from there

Memory persists until you click MC or close the tab. Unlike history (which is read-only), memory accumulates across M+ and M− operations, making it useful for running totals.

Order of Operations

The calculator respects standard mathematical precedence (PEMDAS/BODMAS):

1. Parentheses — innermost first
2. Exponents — right to left: 2^3^2 = 2^(3^2) = 512
3. Functions — applied before multiplication: sin(30) * 2 = 0.5 * 2 = 1
4. Multiplication and Division — left to right
5. Addition and Subtraction — left to right

Use explicit parentheses whenever the precedence isn't obvious. 1 / 2 * pi is (1/2) * pi = 1.5708, not 1 / (2 * pi) = 0.159.

The live preview shows the result of the current expression as you type — this is the fastest way to catch precedence mistakes before committing to equals.

Common Calculation Patterns

Physics — Projectile Range

v₀ = 25 m/s, θ = 40°, g = 9.81 m/s²
Range = v₀² × sin(2θ) / g

25^2 * sin(2*40) / 9.81

Result: 63.5 meters. Remember to set angle mode to DEG.

Finance — Compound Interest

A = P(1 + r/n)^(nt)
P = 10000, r = 0.05, n = 12, t = 10

10000 * (1 + 0.05/12)^(12*10)

Result: 16,470.09 — what $10,000 becomes at 5% compounded monthly for 10 years.

Statistics — Standard Score (Z-score)

z = (x − μ) / σ
x = 78, μ = 70, σ = 8

(78 - 70) / 8

Result: 1. Then look up P(Z < 1) ≈ 0.8413 in a standard normal table — the score is in the 84th percentile.

Decibels

dB = 20 × log₁₀(V₂ / V₁)
V₂/V₁ = 3.162

20 * log10(3.162)

Result: 10 dB — a factor of ~3.16 in voltage amplitude corresponds to +10 dB.

Comparison with Physical Calculators

This calculator targets the overlapping use case: fast expression evaluation, keyboard-friendly workflow, and browser accessibility — without the learning curve of a CAS or the cost of hardware. It replaces a pocket scientific calculator for daily calculation tasks and extends it with history and live feedback.

Privacy and Local Computation

Every calculation runs inside your browser using the mathjs expression engine loaded from the page's JavaScript bundle. No expression is transmitted to any server — there is no backend. This matters in two scenarios:

• Sensitive formulas: financial models, salary calculations, or medical dosing formulas where the numbers themselves are confidential
• Airgapped or restricted environments: the calculator works offline once the page is loaded; no network calls are made during computation

You can verify this yourself: open your browser's DevTools Network tab, evaluate an expression, and confirm that no outbound request appears.

Calculation History

The history panel (toggled via the clock icon) stores up to 50 entries in newest-first order. Each entry shows the original expression and its result. Clicking an entry restores the result to the expression field — not the original expression — so you can continue building on the answer without retyping.

When to use history vs. memory:

• Use memory when you need a value available mid-calculation (intermediate result in a long formula)
• Use history when you want to review, re-derive, or reuse a result from several steps back

History is cleared when you close or refresh the page. If you need to preserve a series of calculations, copy the expressions into a note or spreadsheet.