Jugami Trigonometry

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Trigonometry and Pre-Calculus

Measure what you can’t reach. Then follow the angle all the way around.

This is a free course for high school students in grades 9–12. It starts with the right triangle, the one shape that lets you find a cliff’s height or a river’s width without a tape measure, and builds toward the functions you’ll need for calculus.

Every unit has diagrams that use the same colors for the same sides, step-by-step worked examples, a practice game that checks your answers instantly, and a short quiz.

Start with Right-Triangle TrigonometrySee all units
A unit circle drawn like a surveyor’s compass with tick marks every 10 degrees. A radius of length 1 at angle theta makes a right triangle whose horizontal leg is cos theta and whose vertical leg is sin theta.θcos θsin θ1(cos θ, sin θ)
Put a right triangle inside a circle of radius 1 and its legs are exactly cos θ and sin θ. Keep turning θ past 90° and those two numbers keep working, even with no triangle left. That’s Unit 2.

IntroductionWhat is trigonometry?

The word comes from Greek: trigonon, “triangle,” and metron, “measure.” Trigonometry is the math of how a triangle’s angles and side lengths are tied together.

Here’s the key fact. In a right triangle, once you know one of the sharp angles, the shape is locked in. Big or small, every right triangle with a 35° angle has sides in the same proportions. So if you can measure one angle and one length, you can work out the rest.

That is how surveyors map land, how astronomers first estimated distances in space, and how your phone’s GPS, a game engine, or a sound app turns angles into positions and waves. Pre-calculus then takes these ideas off the triangle and turns them into functions you can graph, which is exactly what calculus needs.

The planUnits

Each unit builds on the one before. Right-Triangle Trigonometry is live now. The others are being written and will appear here when they’re ready.

  1. Live Unit 1

    Right-Triangle Trigonometry

    SOH-CAH-TOA, labeling sides, the 30-60-90 and 45-45-90 triangles, finding missing sides and angles, and angles of elevation and depression. With a triangle explorer, a practice game and a quiz.

  2. Coming soon Unit 2

    The Unit Circle and Radians

    Measuring angles in radians, and defining sine and cosine for any angle, even ones bigger than 90°.

  3. Coming soon Unit 3

    Graphs of Trig Functions

    Sine and cosine waves: amplitude, period, phase shift, and how they model tides, sound and seasons.

  4. Coming soon Unit 4

    Trig Identities

    Equations that are true for every angle, like sin²θ + cos²θ = 1, and how to use them to simplify and solve.

  5. Coming soon Unit 5

    Laws of Sines and Cosines

    Solving triangles that don’t have a right angle, from navigation to bridge trusses.

  6. Coming soon Unit 6

    Functions for Pre-Calculus

    Polynomial, exponential and logarithmic functions, inverses and transformations: the toolkit for calculus.

Try itTools in Unit 1

Before you startWhat you need to know first

You’ll be ready for Unit 1 if you can:

  • solve a one-step equation such as x ÷ 8 = 0.6 or 0.6 = 3 ÷ x;
  • work with square roots, like √9 = 3 and √2 ≈ 1.414;
  • use the Pythagorean theorem, a² + b² = c²;
  • use a scientific calculator (we’ll show you how to check it’s in degree mode).

Need a refresher on that last formula? The Jugami Geometry lesson on the Pythagorean Theorem has proofs, examples and practice.

ColophonAbout these lessons

Every worked example, practice answer and quiz answer is computed twice: once by hand and once by a script that checks the arithmetic. The interactive diagrams are drawn live from the real trig functions, so what you see is what the math says. Practice and quiz best scores are saved only in your own browser.

Credits: all diagrams, icons and the share image are original SVG drawn in code for Jugami, in a “surveyor’s field book” style. Written by Leonhard, Jugami’s trigonometry and pre-calculus teacher.

Sources