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Brilliant

Trigonometry

via Brilliant

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Overview

This course will teach you all of the fundamentals of trigonometry, starting from square one: the basic idea of similar right triangles. In the first sequences in this course, you'll learn the definitions of the most common trigonometric functions from both a geometric and algebraic perspective.

In this course, you'll master trigonometry by solving challenging problems and interacting with animated graphs, not by relying on rote memorization. Additionally, you'll learn how to apply trigonometry in the contexts of measuring and manipulating sound waves and creating complex, artistic designs using polar graphing.
By the end of this course, you’ll be a master at solving triangles, graphing trig functions in both Cartesian and polar coordinates, and applying trigonometric identities.

Syllabus

  • Trigonometry: Learn the key functions of trigonometry.
    • Defining Angles: Before we dive into trigonometry, get comfortable with two ways of measuring angles.
    • Trigonometric Ratios: Explore the ratios of triangles to find trigonometry functions.
    • The Unit Circle: Create a reference tool for finding exact values of trig functions.
  • Measuring: Find the sides and angles of triangles.
    • Solving Right Triangles: Solve for lengths of triangles using trigonometric functions.
    • Solving for Angles: Find the angles of triangles using trigonometric functions.
    • Law of Cosines: Solve for measurements of non-right triangles.
    • Law of Sines: Learn a technique that works when the law of cosines doesn't.
    • Ambiguous Cases: See when measurements make multiple triangles possible, or none.
  • Cartesian Graphing: Explore graphs of trig functions.
    • Trigonometry Graphs: What do the graphs of sine and cosine look like?
    • Other Trigonometric Graphs: Explore some unusual trigonometry graphs, including secant and cosecant.
    • Trigonometric Graph Problem Solving: Creatively apply your skills with trig graphs to solve some challenging puzzles.
    • Trigonometric Graph Symmetry: Apply symmetry as a method of simplification.
    • Inverse Trigonometry: Learn about and apply the inverse trigonometric functions: arcsine, arccosine, and others.
  • Polar Graphing: Plot on a circle rather than a grid.
    • Polar Coordinates: Explore plots that use angle and radius to identify points on the plane.
    • Roses, Cardioids and Limaçon: Discover curves with loops and petals.
    • Function Transformations: Rotate, translate, and dilate polar graphs.
  • Identities: Transform and simplify trigonometry.
    • Reciprocal and Quotient Identities: Rewrite trigonometric expressions by dividing one trig function by another.
    • Triangle Identities: Go back to right triangles and relate trig functions to one another.
    • Sum and Difference Identities: Discover a clever method of handling addition and subtraction inside trig functions.
    • Double- and Half-Angle Identities: Explore scenarios where angle in a trig identity is doubled or halved.

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