Unit 6 of 8

Trigonometric Functions One at a Time

  • 15 lessons
  • Parts 87 to 116

This unit measures angles in radians and reads exact values from the unit circle. It then studies sine, cosine, tangent, cotangent, secant and cosecant one at a time, each with its domain, range, period, zeros and graph.

The fundamental identities and basic trigonometric equations come next, followed by arcsine, arccosine and arctangent with their principal ranges and the compositions of trigonometric and inverse functions. The last lesson rewrites powers and products of sines and cosines, as integration requires.

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 the Unit Circle

  1. Radians and Angle Measurement

    What a radian measures, how to move between radians and degrees, how to find arc length on circles of any size, and how signs and extra full turns describe a rotation.

    Theory part 87Practice part 88

  2. Unit-Circle Coordinates and Exact Values

    How an angle picks out a point on the unit circle, why its coordinates are the cosine and sine, and how two special triangles, reference angles and quadrant signs give exact values.

    Theory part 89Practice part 90

The Six Trigonometric Functions

  1. The Sine Function

    How sine turns every real radian input into the height of a point on the unit circle, and how that height produces the sine wave with its range, period, odd symmetry, zeros and amplitude.

    Theory part 91Practice part 92

  2. The Cosine Function

    How cosine reads the horizontal coordinate of a point on the unit circle, and how that gives its wave with even symmetry, period, zeros, amplitude and midline.

    Theory part 93Practice part 94

  3. The Tangent Function

    How tangent is the ratio of sine to cosine, why it is undefined where cosine is zero, and how its signs, zeros, period and vertical asymptotes shape a graph made of repeating branches.

    Theory part 95Practice part 96

  4. The Cotangent Function

    How cotangent is built as cosine over sine, where it is undefined and where it is zero, and how its period, asymptotes and decreasing branches fit together.

    Theory part 97Practice part 98

  5. The Secant Function

    How secant values come from flipping nonzero cosine heights, which inputs are excluded, why the range skips the band between minus one and one, and how the branches repeat every two pi.

    Theory part 99Practice part 100

  6. The Cosecant Function

    How cosecant comes from flipping nonzero sine heights, how its excluded inputs and odd symmetry differ from secant, and how to sketch one scaled cycle from sine landmarks.

    Theory part 101Practice part 102

Identities and Equations

  1. Trigonometric Identities from the Unit Circle

    How the unit circle gives the Pythagorean, quotient and reciprocal identities, how to use them as replacements, and why every original denominator restriction survives cancellation.

    Theory part 103Practice part 104

  2. Solving Basic Trigonometric Equations

    How to find every solution of a basic sine, cosine or tangent equation in a given interval, including scaled inputs and quadratics in a trigonometric function, and how to check the list.

    Theory part 105Practice part 106

Inverse Trigonometric Functions

  1. The Arcsine Function

    What arcsine returns, why it always picks its angle from one fixed closed interval, and how to find its exact values, domain, range and graph.

    Theory part 107Practice part 108

  2. The Arccosine Function

    Why arccosine reverses the decreasing cosine branch over the upper half of the unit circle, how to find its exact values, and how its domain, range, endpoints and graph follow from that choice.

    Theory part 109Practice part 110

  3. The Arctangent Function

    How arctangent reverses the central tangent branch, why it accepts every real number but never returns a right angle in either direction, and how its graph levels off toward two horizontal asymptotes.

    Theory part 111Practice part 112

  4. Composing Trigonometric and Inverse Functions

    How to evaluate and simplify expressions such as cos(arcsin x) by turning the inverse angle into a point on the unit circle, choosing the sign from its principal interval and keeping the correct domain.

    Theory part 113Practice part 114

Trigonometric Powers and Products

  1. Rewriting Trigonometric Powers and Products

    How double-angle, power-reduction and product-to-sum identities turn squares and products of sines and cosines into sums of single sines and cosines, and how to pick the identity that fits your goal.

    Theory part 115Practice part 116