sin, cos, tan and their inverses
For an angle θ in a right-angled triangle, sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse and tan θ = opposite/adjacent. More generally, if you go round a circle of radius 1 by angle θ, cos θ and sin θ are the x and y coordinates of where you end up, which is why they work for any angle, including negative ones and ones over 360°.
| Key | Accepts | Returns | Example (DEG) |
|---|---|---|---|
| sin | any angle | −1 to 1 | sin(30) = 0.5 |
| cos | any angle | −1 to 1 | cos(60) = 0.5 |
| tan | any angle except 90° ± 180°k | any number | tan(45) = 1 |
| sin⁻¹ | −1 to 1 | −90° to 90° | sin⁻¹(0.5) = 30 |
| cos⁻¹ | −1 to 1 | 0° to 180° | cos⁻¹(0) = 90 |
| tan⁻¹ | any number | between −90° and 90° | tan⁻¹(1) = 45 |
The inverse keys can only return one answer, the "principal value", even though many angles share the same sine. sin 30° and sin 150° are both 0.5, but sin⁻¹(0.5) gives 30. If your problem needs the other one, you have to work it out yourself; the triangle guide shows where this matters.
Whether angles are measured in degrees or radians depends on the DEG/RAD button. Getting this wrong is the single most common calculator mistake, so it has its own guide.
Hyperbolic functions (HYP)
Press HYP and the sin, cos and tan keys become sinh, cosh and tanh. Despite the names they have nothing to do with angles; they're combinations of exponentials:
sinh x = (eˣ − e⁻ˣ) / 2 cosh x = (eˣ + e⁻ˣ) / 2 tanh x = sinh x / cosh x
They earn the "trig" names because they satisfy similar identities (cosh²x − sinh²x = 1, compared with cos²x + sin²x = 1). Where they turn up:
- cosh is the shape of a hanging chain or cable, the catenary y = a cosh(x/a).
- tanh describes how velocity approaches terminal velocity under air resistance, and is used as an activation function in neural networks.
- sinh and cosh appear in heat flow along fins, transmission-line equations and special relativity.
The DEG/RAD setting doesn't affect them. Example: cosh(1) = 1.54308063482.
ln, log, eˣ and 10ˣ
A logarithm answers "what power do I raise the base to, to get this number?" log uses base 10, so log 1000 = 3 because 10³ = 1000. ln uses base e ≈ 2.71828, so ln e = 1. The keys 10ˣ and eˣ undo them.
- Both only accept positive numbers. log 0 and the log of a negative number have no real value, and Kalkuli says so.
- For any other base, use change of base: log₂ 8 =
ln(8) ÷ ln(2)= 3. - ln is the "natural" one because the rate of growth of eˣ is eˣ itself, which makes it the right base for anything that grows or decays in proportion to its size: interest, populations, radioactive decay, capacitor discharge.
Base 10 is used for scales that span many orders of magnitude, like decibels and pH. Worked examples are in logarithms in real life.
Powers, roots and reciprocals
| Key | Types | Example |
|---|---|---|
| x², x³ | ², ³ after the number | (3+4)² = 49 |
| xʸ | ^ | 2^10 = 1024, 8^(1/3) = 2 |
| √x, ∛x | √(, ∛( | √(144) = 12, ∛(-27) = −3 |
| 1/x | ⁻¹ after the number | 8⁻¹ = 0.125 |
Powers are evaluated right to left, so 2^3^2 = 2⁹ = 512, not 8² = 64. That's the standard mathematical convention, though some spreadsheets do it the other way.
A negative number raised to a fractional power usually isn't a real number: (−8)^0.5 would be √−8. Kalkuli makes one exception that matches what you'd expect from handwritten maths: odd roots like (-8)^(1/3) return −2.
Factorial, absolute value and Rand
n! multiplies all the whole numbers from 1 to n: 5! = 5 × 4 × 3 × 2 × 1 = 120, and 0! = 1 by definition. Factorials grow astonishingly fast. 170! ≈ 7.3 × 10³⁰⁶ is the largest that fits in a standard floating-point number; ask for 171! and Kalkuli will tell you it's too large. Only whole numbers are accepted.
|x| gives the size of a number ignoring its sign: abs(-7.5) = 7.5.
Rand inserts a random number between 0 and 1 (to six decimal places). Rand × 6 rounded up simulates a dice roll.
Memory keys
- MS stores the current value, replacing whatever was in memory.
- M+ and M− add the current value to memory, or subtract it.
- MR inserts the stored value into your expression.
- MC clears memory back to zero.
An M badge appears next to DEG/RAD whenever memory isn't zero. Memory is handy for running totals: work out each item, press M+, and press MR at the end. If the expression on screen isn't finished yet, the memory keys won't guess; they'll ask you to complete it first.
Order of operations
Kalkuli follows the standard order, often remembered as BIDMAS or PEMDAS:
- Brackets, innermost first.
- Indices: powers and roots, including ², ³, !, % and ⁻¹ written after a number.
- Division and multiplication, left to right. Implicit multiplication like
2πcounts as multiplication. - Addition and subtraction, left to right.
6 ÷ 2(1 + 2) is famous for splitting opinion. Kalkuli treats implicit multiplication the same as ×, and works left to right, giving (6 ÷ 2) × 3 = 9. Some textbooks and older calculators bind 2(1 + 2) more tightly and get 1. If there's any doubt, add brackets: 6 ÷ (2(1 + 2)).