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Trigonometry
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.
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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)
Chapters from OpenStax Algebra and Trigonometry 2e
Every section of the book, condensed into a lesson with its own practice problems.
8. Periodic Functions
Introduction to Periodic FunctionsGraphs of the Sine and Cosine FunctionsGraphs of the Other Trigonometric FunctionsInverse Trigonometric Functions
9. Trigonometric Identities and Equations
Chapters from OpenStax Precalculus 2e
Every section of the book, condensed into a lesson with its own practice problems.
5. Trigonometric Functions
Introduction to Trigonometric FunctionsUnit Circle: Sine and Cosine FunctionsThe Other Trigonometric FunctionsRight Triangle Trigonometry
7. Trigonometric Identities and Equations
Introduction 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
8. 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
Symbols used here
Ratio of a circle's circumference to its diameter, 3.14159…
The usual name for an angle.
1/360 of a full turn. 180° = π radians.
Ratios of sides in a right triangle; coordinates on the unit circle.
The angle whose sine is the given value (and likewise arccos, arctan).
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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