1. Trigonometric & Circular Functions
Trigonometric functions describe geometric relationships between angles and side lengths in right triangles, and extend to periodic motion on the Cartesian coordinate plane via the unit circle (a circle of radius r = 1 centered at the origin).
Sine (sin θ) & Inverse Sine (arcsin / sin⁻¹)
sin(θ) = Opposite / Hypotenuse = y / r
The sine function oscillates between -1 and +1 with a fundamental period of 2π radians (360°). The inverse function arcsin(x) calculates the angle whose sine is x, defined strictly for domain [-1, 1] with range [-π/2, π/2] (or [-90°, +90°]).
Cosine (cos θ) & Inverse Cosine (arccos / cos⁻¹)
cos(θ) = Adjacent / Hypotenuse = x / r
Cosine represents the horizontal projection of a point on the unit circle. It is an even function satisfying cos(-θ) = cos(θ). The inverse function arccos(x) is defined for x ∈ [-1, 1] with principal value range [0, π] (or [0°, 180°]).
Tangent (tan θ) & Inverse Tangent (arctan / tan⁻¹)
tan(θ) = sin(θ) / cos(θ) = y / x
Tangent measures the slope of the radial line. Tangent possesses vertical asymptotes wherever cos(θ) = 0 (e.g., at ±90°, ±270°). The inverse function arctan(x) accepts any real number (-∞, +∞) and outputs angles in the range (-π/2, π/2).
2. Hyperbolic Trigonometry (HYP Mode)
While standard trigonometric functions parametrize the unit circle x² + y² = 1, hyperbolic functions parametrize the unit equilateral hyperbola x² - y² = 1. They arise extensively in civil engineering (catenary suspension cables), thermodynamics, special relativity (Lorentz boosts), and transmission line theory.
| Function | Exponential Definition | Domain | Geometric / Physical Significance |
|---|---|---|---|
| Hyperbolic Sine (sinh x) | (eˣ - e⁻ˣ) / 2 |
(-∞, +∞) | Odd function; models fluid flow velocity profiles. |
| Hyperbolic Cosine (cosh x) | (eˣ + e⁻ˣ) / 2 |
(-∞, +∞) | Even function; describes the catenary curve of hanging power cables. |
| Hyperbolic Tangent (tanh x) | sinh(x) / cosh(x) = (e²ˣ - 1) / (e²ˣ + 1) |
(-∞, +∞) | Activation function in machine learning; relativistic velocity addition. |
Usage Tip: To evaluate hyperbolic functions on the calculator, click the HYP toggle button above the display, then click sin, cos, or tan.
3. Logarithms & Exponential Scaling
Logarithms represent the inverse operation to exponentiation. The logarithm of a number x to a given base b is the power to which b must be raised to produce x: log_b(x) = y ⟺ bʸ = x.
Natural Logarithm (ln x) & eˣ
Uses Napier's constant / Euler's number e ≈ 2.718281828459. The natural logarithm is essential in differential calculus because the derivative of ln(x) is simply 1/x, and the derivative of eˣ is eˣ.
Common Logarithm (log x) & 10ˣ
Uses base 10. The common logarithm is standard in engineering and measurement scales where physical magnitudes span multiple orders of magnitude:
- Richter Magnitude: Measures seismic wave amplitude.
- pH Chemistry Scale:
pH = -log₁₀[H⁺] - Decibels (dB): Audio and signal attenuation.
4. Powers, Roots & Combinatorial Factorials
Power and root operations evaluate nonlinear relationships in physics, geometric volumes, and standard deviation in statistics:
Arbitrary Powers (xʸ) & Roots (√x, ∛x)
The xʸ key raises base x to exponent y. Square root (√x) and cube root (∛x) represent fractional exponents x^(1/2) and x^(1/3) respectively. Square roots of negative real numbers return an error alert in real-number mode.
Factorial (n!) & Combinatorics
The factorial operation n! calculates the product of all positive integers less than or equal to n. In combinatorics, factorials quantify the number of possible orderings of n distinct objects. By mathematical definition, 0! = 1.
5. Order of Operations & Expression Hierarchy
Mathematical expressions must be evaluated according to standard algebraic hierarchy (often remembered by the acronyms PEMDAS or BODMAS):
- Parentheses / Brackets: Expressions enclosed in
( ... )are computed first. Deeply nested parentheses evaluate from the innermost pair outward. - Exponents & Roots: Powers (
xʸ,x²,x³) and radicals (√,∛). - Multiplication & Division: Evaluated in strict left-to-right order (
×,÷). - Addition & Subtraction: Evaluated in left-to-right order (
+,−).