Trigonometry Formulas

De Moivre's Theorem

Reference for De Moivre's theorem for raising complex numbers to any power and finding nth roots. Includes proof outline and worked examples with unit circle.

Napier's Analogies

Reference for Napier's analogies for solving oblique spherical triangles. All four proportional formulas with variable definitions and step-by-step examples.

Triple Angle Formulas

Triple angle identities: sin(3x) = 3sin(x) - 4sin³(x) and cos(3x) = 4cos³(x) - 3cos(x). Derived step-by-step with worked examples and real applications.

Versine Formula

Calculate the versine of an angle, a classic trigonometric function used in navigation, surveying, and historical computation tables.

Weierstrass Substitution

Reference for the Weierstrass substitution t = tan(x/2) that converts trigonometric integrals into rational functions. With full derivation and worked examples.

Cofunction Identities

Reference for the six cofunction identities: sin/cos, tan/cot, and sec/csc. Includes proofs and applications for simplifying trigonometric expressions.

Hyperbolic Functions

Definitions and properties of sinh, cosh, and tanh hyperbolic functions. Learn their formulas, identities, and applications with examples.

Inverse Trig Derivatives

Reference for derivatives of inverse trig functions: arcsin, arccos, arctan, arccot, arcsec, and arccsc. Includes domain restrictions and worked examples.

Inverse Trigonometric Functions

Domains, ranges, and key properties of arcsin, arccos, and arctan. Learn inverse trig functions with worked examples and identities.

Mollweide's Formula

Mollweide's formula relates sides and angles of a triangle in a single equation. Useful for checking triangle solutions with examples.

Polar to Cartesian Conversion

Convert between polar coordinates (r, theta) and Cartesian coordinates (x, y). Includes formulas for both directions with examples.

Power Reduction Formulas

Reference for power reduction formulas that rewrite sin2(x) and cos2(x) as first-power trig expressions. Essential for integration and Fourier analysis.

Prosthaphaeresis Formulas

Prosthaphaeresis formulas convert products of trig functions into sums. Learn these historical formulas with step-by-step examples.

Reciprocal Identities

Reference for the six reciprocal trig identities: sec, csc, and cot as reciprocals of cos, sin, and tan. Includes definitions and domain restrictions.

Secant, Cosecant, and Cotangent Formulas

Reciprocal trig functions: sec = 1/cos, csc = 1/sin, cot = 1/tan. Covers definitions, Pythagorean identities sec²−tan²=1 and csc²−cot²=1, with worked examples.

Angle Addition Formulas

Reference for sin(A±B), cos(A±B), and tan(A±B) angle addition formulas. Covers unit circle derivation and double-angle and half-angle identity applications.

Law of Tangents

Law of tangents formula (a−b)/(a+b) = tan[(A−B)/2] / tan[(A+B)/2] solves triangles when two sides and the included angle are known. Alternative to cosine rule.

Product-to-Sum Formulas

Reference for product-to-sum trig identities for sin(A)cos(B), cos(A)cos(B), and sin(A)sin(B) with sum-to-product formulas and worked examples.

Radian Measure and Arc Length

Convert between degrees and radians and calculate arc length from radian measure. Covers the unit circle, degree-to-radian conversion, and worked examples.

Sum-to-Product Formulas

Convert sums of trig functions into products. Simplify expressions and solve equations in trigonometry and signal processing.

Area of a Triangle (Trigonometric)

Calculate triangle area using A = ½ab·sin(C) when two sides and the included angle are known. Covers the trig formula alongside Heron and base-height methods.

Double Angle Formulas

Reference for double angle formulas: sin(2θ)=2sinθcosθ, cos(2θ)=cos²θ-sin²θ, and tan(2θ). Includes worked examples and derivations from the angle sum formulas.

Half Angle Formulas

Reference for half-angle formulas for sin, cos, and tan. Derived from double-angle identities for integration, exact values, and simplifying trig expressions.

Heron's Formula

Calculate the area of any triangle from its three side lengths using Heron's formula: A = √(s(s-a)(s-b)(s-c)). No angles needed.

Inverse Trigonometric Functions

Definitions and ranges for arcsin, arccos, and arctan. Find angles from trig values with inverse trig functions and worked examples.

Law of Cosines

The Law of Cosines c² = a² + b² - 2ab·cos(C) solves any triangle when you know three sides or two sides and the included angle.

Law of Sines

The Law of Sines relates sides and angles of any triangle: a/sin(A) = b/sin(B) = c/sin(C). Solve triangles with step-by-step examples.

Pythagorean Theorem

The Pythagorean theorem a² + b² = c² finds the length of any side of a right triangle. The most famous formula in geometry.

SOHCAHTOA

SOHCAHTOA is the mnemonic for the three basic trig ratios: sin = O/H, cos = A/H, tan = O/A. Essential for solving right triangles.

Trigonometric Identities

Essential trig identities: Pythagorean identities, reciprocal identities, and quotient identities. A complete reference for simplifying trig expressions.

Unit Circle Values

Complete reference for sine, cosine, and tangent values at key angles (0°, 30°, 45°, 60°, 90° and beyond). Essential trig reference table.