Unit 4 of 10

Trigonometric Functions

  • 11 lessons
  • Parts 93 to 114

Trigonometry begins with angles and triangles and becomes the study of functions that repeat. This unit measures angles in degrees and radians, defines the six trigonometric ratios in right triangles, and extends them to every angle with reference angles and the unit circle.

From the unit circle come the graphs of sine and cosine, and then amplitude, period and phase shift for fitting waves to periodic data. The unit also graphs tangent, cotangent, secant and cosecant, and ends with harmonic motion, the model for springs and other oscillations.

Every lesson will be a pair of videos, a theory part that explains the idea and a practice part that works through problems. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.

Angles and Arcs

  1. Angles, Radians, and Coterminal Angles

    How to place an angle in standard position, measure it in degrees or in radians, convert between the two, and recognize coterminal angles that share a terminal side.

    Theory part 93Practice part 94

  2. Arc Length and Area of a Sector

    How a central angle in radians gives the length of an arc and the area of a sector, and how the turning rate of a wheel sets the speed of a point on its rim.

    Theory part 95Practice part 96

Triangles and the Unit Circle

  1. Right Triangle Trigonometry

    How the sides of a right triangle define sine, cosine, tangent and their reciprocals, why these ratios depend only on the angle, and how to use them to find heights and distances.

    Theory part 97Practice part 98

  2. Special Right Triangles

    The exact side ratios of the 45-45-90 and 30-60-90 triangles, how they give exact trigonometric values at 30, 45 and 60 degrees, and how to rationalize a denominator.

    Theory part 99Practice part 100

  3. General Angles and Reference Angles

    How a point on the terminal side defines all six trigonometric functions for any angle, where each one is undefined, and how reference angles and quadrant signs give exact values.

    Theory part 101Practice part 102

  4. The Unit Circle

    How the unit circle turns every real number into a point whose coordinates are its cosine and sine, why these values repeat after every full turn, and how symmetry gives exact values all around the circle.

    Theory part 103Practice part 104

Graphs of Trigonometric Functions

  1. Graphs of Sine and Cosine

    How the sine and cosine waves come from a point moving around the unit circle, and how to read their period, amplitude and midline from a graph or a formula.

    Theory part 105Practice part 106

  2. Amplitude, Period, and Phase Shift

    What each of the four constants in a general sine or cosine function does to the graph, and how to find them from a graph or from data that rise and fall.

    Theory part 107Practice part 108

  3. Graphs of Tangent and Cotangent

    Why the tangent and cotangent graphs break into branches separated by vertical asymptotes, why they repeat twice as often as sine and cosine, and how to sketch them from a few key points.

    Theory part 109Practice part 110

  4. Graphs of Secant and Cosecant

    How the secant and cosecant graphs come from taking reciprocals of cosine and sine heights, why they split into U-shaped branches, and why they never take a value strictly between minus one and one.

    Theory part 111Practice part 112

Oscillation

  1. Harmonic Motion

    How a sine or cosine function describes a weight bouncing on a spring, what amplitude, period and phase mean for that motion, and how an exponential factor models swings that fade.

    Theory part 113Practice part 114