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Trigonometry

Angles, the unit circle, identities and trig equations.

Lessons

Introductory The unit circle Exact values of sine and cosine at the special angles. sin(pi/3) Core Trigonometric equations Reference angles and the infinitely many solutions. sin(x) = 1/2 Core Trigonometric identities Pythagorean, double-angle and sum identities, verified by simplification. simplify sin(x)^2 + cos(x)^2 Introductory Degrees and radians Two ways to measure an angle, and converting between them. 30 degrees to radians Core Right-triangle trigonometry (SOH-CAH-TOA) Sine, cosine and tangent as side ratios, and solving right triangles. hypotenuse of 5 and 12 Core Law of sines and law of cosines Solving any triangle from three pieces of information. triangle 5 7 with angle 60 Core Graphs of sine, cosine and tangent Amplitude, period, phase shift and the shape of periodic motion. y = sin(x) + cos(2x) Core Inverse trigonometric functions arcsin, arccos, arctan (which angle has this ratio) and their derivatives. derivative of arctan(x)

Trigonometry is the geometry of the circle written as functions. Exact values from the unit circle, equations with infinitely many solutions, and the identities that make them tame.

More lessons

Trigonometric Identities and Equations

Verifying Trigonometric Identities and Using Trigonometric Identities to Simplify Trigonometric ExpressionsIntroduction to Trigonometric Identities and EquationsSimplifying and Verifying Trigonometric IdentitiesSum and Difference IdentitiesDouble-Angle, Half-Angle, and Reduction FormulasSum-to-Product and Product-to-Sum FormulasSolving Trigonometric EquationsModeling with Trigonometric Functions

Trigonometric Functions

Introduction to Trigonometric FunctionsUnit Circle: Sine and Cosine FunctionsThe Other Trigonometric FunctionsRight Triangle Trigonometry

Periodic Functions

Introduction to Periodic FunctionsGraphs of the Sine and Cosine FunctionsGraphs of the Other Trigonometric Functions

Further Applications of Trigonometry

Introduction to Further Applications of TrigonometryNon-right Triangles: Law of SinesNon-right Triangles: Law of CosinesPolar CoordinatesPolar Coordinates: GraphsPolar Form of Complex NumbersParametric EquationsParametric Equations: GraphsVectors

The functions

TrigonometryTrigonometric functionsRadianSine and cosineTangent (trigonometry)Inverse trigonometric functionsHyperbolic functions

Identities and triangles

List of trigonometric identitiesPythagorean trigonometric identityLaw of sinesLaw of cosinesLaw of tangentsSolution of trianglesAngle addition formulaDouble-angle formulaSmall-angle approximation

Some of these lessons are adapted from open texts: OpenStax Algebra and Trigonometry 2e, OpenStax Precalculus 2e, Wikipedia. Each lesson page names its source and licence.

Symbols used here

Tap any symbol for the full definition, a picture, and what every letter in it means.

Questions people ask

Why radians instead of degrees?

A radian is the angle whose arc equals the radius, so it is a pure ratio rather than an arbitrary 1/360 of a turn. With radians the derivative of sin x is exactly cos x; with degrees a factor of π/180 appears everywhere.

Why does sin x = 1/2 have infinitely many solutions?

Sine repeats every full turn, and within one turn it reaches 1/2 twice (at π/6 and 5π/6). Add any whole number of turns to either and it is still true.

How do I remember the exact values?

Two triangles: the 45-45-90 with sides 1, 1, √2 and the 30-60-90 with sides 1, √3, 2. Every value for 30°, 45° and 60° is a ratio of those sides; symmetry gives the rest of the circle.

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