Topics in Trigonometry

Topics in Trigonometry

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Definitions, Values, and Identities:

The Definition(s) of Trigonometry: the unit circle, standard position ratios, and triangle ratios
Basic Trigonometric Values: multiples of 30 and 45 degrees
Multiples of 15 Degrees: derivations of trig values for 15 and 75 degrees
Multiples of 18 Degrees: derivations of trig values for 18, 36, 54, and 72 degrees
Trig Values and the Unit Circle (webMathematica): vary the parameter to see the functions as projections
Basic Trigonometric Identities: reciprocal, ratio, Pythagorean, symmetric, and cofunction identities
Angle Sum and Difference Identities: angle sum, angle difference, double angle, power-reducing, half-angle, product-to-sum, and sum-to-product formulas
Triangle Identities: Law of Sines, Law of Cosines, Law of Tangents, and a product-sum identity
Definitions and Graphs of the Inverse Trig Functions: the six inverse trig functions (some with two forms)
Basic Inverse Trig Function Identities: reciprocal, symmetric, and cofunction identities

Graphs:

Trigonometric Graphs: the six basic graphs understood as projections
Generating the Sine Function (GIF animation): see the sine function developed from the unit circle
The Trig Functions and the Unit Circle (GIF animation): see the 6 trig functions vary through the first quadrant of the unit circle
Exploring the Graph of the Sine Function (Excel): vary the coefficients of the sine function and watch the graph change
Exploring the Graph of the Cosine Function (Excel): vary the coefficients of the sine function and watch the graph change
Exploring the Graph of the Tangent Function (Excel): vary the coefficients of the sine function and watch the graph change
Exploring the Graph of the Cotangent Function (Excel): vary the coefficients of the sine function and watch the graph change
Exploring the Graph of the Secant Function (Excel): vary the coefficients of the sine function and watch the graph change
Exploring the Graph of the Cosecant Function (Excel): vary the coefficients of the sine function and watch the graph change
Curvilinear Coordinates and the Sine Function (Excel): vary the coefficients of the sine function and watch the graph in rectangular, polar, parabolic, elliptic, or bipolar coordinates
Curvilinear Coordinates and the Tangent Function (Excel): vary the coefficients of the tangent function and watch the graph in rectangular, polar, parabolic, elliptic, or bipolar coordinates
Curvilinear Coordinates and the Secant Function (Excel): vary the coefficients of the secant function and watch the graph in rectangular, polar, parabolic, elliptic, or bipolar coordinates
Definitions and Graphs of the Inverse Trig Functions: the six inverse trig functions (some with two forms)
Compositions of Trig and Inverse Trig Functions: 108 compositions with 74 graphs
Curvilinear Coordinates and the Inverse Sine Function (Excel): vary the coefficients of the inverse sine function and watch the graph in rectangular, polar, parabolic, elliptic, or bipolar coordinates
Curvilinear Coordinates and the Inverse Tangent Function (Excel): vary the coefficients of the inverse tangent function and watch the graph in rectangular, polar, parabolic, elliptic, or bipolar coordinates
Eccentricity of a Conic (GIF animation): see the conic graph change as its eccentricity varies
Rose Analyzer (webMathematica): vary the parameters in the polar equation of the rose
Limacon Analyzer (webMathematica): vary the parameters in the polar equation of the limacon
Lissajous Curve Analyzer (webMathematica): vary the parameters in the parametric equations of family of curves of Lissajous

Complex Trigonometry:

Imaginary Numbers and Trigonometry: understanding non-real numbers using trig
Functions of Complex Numbers: defining trig and inverse trig functions for non-real values
Complex Sine Function (webMathematica): vary the parameters in the complex sine function, and view subsets of the 4D graph
Complex Arcsine Function (webMathematica): vary the parameters in the complex arcsine function, and view subsets of the 4D graph
Complex Exponential Function (webMathematica): vary the parameters in the complex exponential function, and view subsets of the 4D graph (especially the imaginary parts)

Matrices and Series:

2D Matrix Transformations (webMathematica): input a matrix and see it iteratively transform a simple geometric object
2D Homogeneous Transformations (webMathematica): input a matrix and see it iteratively transform a simple geometric object represented with homogeneous coordinates
Exploring Fourier Series (webMathematica): vary the generic Fourier series term, and see the effect on the graph
Taylor Polynomials (GIF animation): see the Taylor polynomials converge to the sine function

Drillmasters:

Standard Position Trig Ratios Drillmaster (webMathematica): practice the 6 basic trig ratios from an angle in standard position
Right Triangle Trig Ratios Drillmaster (webMathematica): practice the 6 basic trig ratios from a right triangle
Converting Radians to Degrees Drillmaster (webMathematica): practice converting special angles to degrees
Converting Degrees to Radians Drillmaster (webMathematica): practice converting special angles to radians
Special Angle Sine and Cosine Values Drillmaster (webMathematica): practice the sine and cosine ratios for the special angles, in both degrees and radians
Special Angle Trig Values Drillmaster (webMathematica): practice the 6 basic trig ratios for the special angles, in both degrees and radians
Sine Graph Characteristics Drillmaster (webMathematica): practice identifying amplitude, period, and vertical translation from a graph