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LMAT 106 · Unit 1
Interactive mathematics course

Angles and the Unit Circle

Build every idea from the beginning, then use it to evaluate sine and cosine exactly. The 21 topics follow one connected learning path, with visuals, fully explained examples, guided practice, independent practice, and feedback.

How to use this course

Follow the topics in order

Each topic teaches one piece of the mathematics and prepares you for the next. Start with Topic 1 if angles or the unit circle are new to you. Return here at any time to reopen a topic or continue from the last one you visited.

Part 1

Angle foundations

  1. What an Angle RepresentsBuild the idea of an angle as a directed rotation between two rays. Open topic
  2. Standard PositionPlace an angle at the origin with its initial side on the positive x-axis. Open topic
  3. Positive and Negative RotationUse counterclockwise and clockwise motion to determine the sign of an angle. Open topic
  4. Quadrants and Quadrantal AnglesLocate terminal sides in quadrants and recognize angles that end on an axis. Open topic
  5. Coterminal AnglesGenerate and interpret angles that finish on the same terminal side. Open topic
Part 2

Radians and conversions

  1. Circle Vocabulary and Arc LengthConnect radius, circumference, central angle, and intercepted arc length. Open topic
  2. What a Radian RepresentsUnderstand one radian through the relationship between radius and arc length. Open topic
  3. Degree and Radian LandmarksLearn the degree and radian measures for the circle's major landmarks. Open topic
  4. Converting Degrees to RadiansConvert degree measures exactly and simplify the resulting fractions of pi. Open topic
  5. Converting Radians to DegreesConvert radian measures to degrees by canceling units deliberately. Open topic
Part 3

Building the unit circle

  1. Unit-Circle CoordinatesConnect an angle's terminal point with its ordered pair on the unit circle. Open topic
  2. The Pythagorean RelationshipUse x² + y² = 1 to reason about every point on the unit circle. Open topic
  3. Deriving 45° CoordinatesDerive the exact 45° unit-circle coordinates from an isosceles right triangle. Open topic
  4. Deriving 30° and 60° CoordinatesDerive exact coordinates from a 30-60-90 triangle instead of memorizing them blindly. Open topic
  5. Reference AnglesFind the acute angle between a terminal side and the x-axis in every quadrant. Open topic
  6. Coordinate Signs by QuadrantDetermine the signs of x and y from the terminal side's quadrant. Open topic
Part 4

Exact trigonometric values

  1. Defining Cosine as the x-CoordinateRead cosine as the horizontal coordinate of a unit-circle point. Open topic
  2. Defining Sine as the y-CoordinateRead sine as the vertical coordinate of a unit-circle point. Open topic
  3. Evaluating Sine ExactlyCombine reference angles and quadrant signs to evaluate sine exactly. Open topic
  4. Evaluating Cosine ExactlyCombine reference angles and quadrant signs to evaluate cosine exactly. Open topic
  5. Evaluating Multi-Turn and Radian AnglesReduce multi-turn degree and radian angles before evaluating sine or cosine. Open topic