🔬 Full Scientific Calculator
Function Reference Guide
Every button on this tool corresponds to a defined mathematical operation. The guide below explains each category, including the domain constraints that determine when an operation is valid.
Trigonometric Functions
sin cos tan compute the sine, cosine, and tangent of an angle. Switch between degree and radian mode using the DEG/RAD toggle. Note: tan(90°) is undefined — the result approaches infinity.
Inverse Trigonometric Functions
sin⁻¹ cos⁻¹ tan⁻¹ return the angle whose trig value equals the input. Domain: sin⁻¹ and cos⁻¹ require input in [−1, 1]; tan⁻¹ accepts all real numbers. Output is in degrees or radians depending on mode.
Logarithms
log computes the base-10 logarithm — the exponent to which 10 must be raised to produce the input. ln is the natural logarithm (base e). Both require strictly positive inputs; log(0) and ln(0) are undefined (−∞).
Powers and Roots
x² squares the current value. x³ cubes it. xⁿ raises it to any power you enter. √x is the square root (requires x ≥ 0). ∛x is the cube root, which accepts negative inputs.
Factorial (n!)
Computes the product of all positive integers up to n. Defined only for non-negative integers: 0! = 1, 5! = 120, 10! = 3,628,800. Growth is extremely rapid — values above 170! exceed floating-point range.
Memory Functions
M+ adds the current display value to memory. M− subtracts it. MR recalls the stored value to the display. MC clears memory to zero. Useful for multi-step calculations where an intermediate result must be retained.
Mathematical Constants
π (pi) = 3.14159265358979... — the ratio of a circle's circumference to its diameter. e (Euler's number) = 2.71828182845904... — the base of the natural logarithm and the foundation of exponential growth models.
Angle Modes: DEG vs RAD
In degree mode, a full rotation equals 360°. In radian mode, a full rotation equals 2π ≈ 6.2832 rad. The conversion formula is: radians = degrees × π ÷ 180. Physics and advanced mathematics typically use radians; everyday geometry typically uses degrees.
Where These Functions Appear in Academic Study
Advanced mathematical functions are foundational across a range of university disciplines. Understanding their application — not just how to press the button — improves both exam performance and practical problem-solving.
Physics — Wave Mechanics
Simple harmonic motion, electromagnetic waves, and AC circuit analysis all rely on sinusoidal functions. Displacement in SHM: x = A·sin(ωt + φ), where ω (angular frequency) relates directly to radians per second.
Biology & Ecology — Growth Rates
Population dynamics use the natural logarithm to linearise exponential growth data. Doubling time T₂ = ln(2) ÷ r, where r is the intrinsic growth rate. Radioactive decay half-life uses the same relationship: t½ = ln(2) ÷ λ.
Chemistry — pH and Equilibrium
pH = −log₁₀[H⁺]. The Henderson-Hasselbalch equation uses base-10 logarithms to calculate buffer pH. Equilibrium constant expressions in thermodynamics use both log and ln depending on the form of the Gibbs free energy equation.
Engineering — Signal Processing
Decibels (dB) are defined as 10·log₁₀(P₂/P₁) for power ratios. Fourier transforms, which decompose signals into frequency components, rely on complex exponentials e^(iωt). Transfer functions in control systems use complex logarithms.
Statistics — Probability Distributions
The normal distribution's probability density function contains e raised to a power. The Poisson distribution uses e. Logistic regression in data science uses the natural log in the log-odds (logit) transform.
Economics & Finance
Continuous compound interest: A = Pe^(rt). The Black-Scholes option pricing model requires both exponential and logarithmic functions. GDP growth rate analysis uses percentage change; long-run trend analysis uses logarithmic transformation to stabilise variance.
Quick Reference: Input, Operation, and Expected Output
| Button | Mathematical Operation | Example Input | Example Output |
|---|---|---|---|
| sin (DEG) | Sine of angle in degrees | 30 | 0.5 |
| cos (DEG) | Cosine of angle in degrees | 60 | 0.5 |
| tan (DEG) | Tangent of angle in degrees | 45 | 1 |
| sin⁻¹ (DEG) | Inverse sine — returns degrees | 0.5 | 30 |
| log | Base-10 logarithm | 1000 | 3 |
| ln | Natural logarithm (base e) | e (2.71828) | 1 |
| √x | Square root | 144 | 12 |
| ∛x | Cube root | 27 | 3 |
| x² | Square | 13 | 169 |
| xⁿ | Arbitrary power | 2, n=10 | 1024 |
| n! | Factorial | 6 | 720 |
| π | Inserts constant π | — | 3.14159… |
| e | Inserts Euler's number | — | 2.71828… |