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What you will learn in Trigonometry

1. Ratios in a Right Triangle
Similar right triangles have proportional sides, which is the only reason a ratio can be named after an angle.
2. Exact Values from Two Triangles
The half-equilateral and the half-square generate every exact ratio a learner needs, which is why the triangles are worth more than the table.
3. The Unit Circle and Any Angle
Extending the ratios beyond 90° by putting the triangle inside a circle, where the signs come from which quadrant the point is in.
4. Radian Measure as the Analytic Unit
One radian is the angle whose arc equals the radius — a geometric fact rather than a conversion factor, and the unit calculus requires.
5. Graphs of Sine, Cosine and Tangent
The wave is the circle unrolled: the same point's height, plotted against the angle it has turned through.
6. Pythagorean and Reciprocal Identities
One identity, proved on the unit circle, and the two others that are it rearranged.
7. Compound-Angle Formulas
Built from a construction, not quoted — because every other formula in the chapter descends from it.
8. Double- and Half-Angle Formulas
Every one of them is the compound formula with φ = θ. There is nothing else in them.
9. Sum-to-Product and Product-to-Sum
The identities behind beats and interference — two waves added giving one wave inside an envelope.
10. Trigonometric Equations
Infinitely many solutions, and an interval that makes the list finite. Finding one is the easy half.
11. Inverse Trigonometric Functions
Sine is not one-to-one, so it has no inverse — until you cut its domain down until it is.
12. Sine Rule, Cosine Rule and Area
Triangles that are not right-angled — and the case where two different triangles fit the same data.
13. Heights and Distances
Angles of elevation and depression, bearings, and the diagram that does most of the work.
14. Harmonic Form
a sin θ + b cos θ is a single wave. That fact is what superposition means.

What you will learn in Trigonometry

1. Ratios in a Right Triangle

Similar right triangles have proportional sides, which is the only reason a ratio can be named after an angle.

2. Exact Values from Two Triangles

The half-equilateral and the half-square generate every exact ratio a learner needs, which is why the triangles are worth more than the table.

3. The Unit Circle and Any Angle

Extending the ratios beyond 90° by putting the triangle inside a circle, where the signs come from which quadrant the point is in.

4. Radian Measure as the Analytic Unit

One radian is the angle whose arc equals the radius — a geometric fact rather than a conversion factor, and the unit calculus requires.

5. Graphs of Sine, Cosine and Tangent

The wave is the circle unrolled: the same point's height, plotted against the angle it has turned through.

6. Pythagorean and Reciprocal Identities

One identity, proved on the unit circle, and the two others that are it rearranged.

7. Compound-Angle Formulas

Built from a construction, not quoted — because every other formula in the chapter descends from it.

8. Double- and Half-Angle Formulas

Every one of them is the compound formula with φ = θ. There is nothing else in them.

9. Sum-to-Product and Product-to-Sum

The identities behind beats and interference — two waves added giving one wave inside an envelope.

10. Trigonometric Equations

Infinitely many solutions, and an interval that makes the list finite. Finding one is the easy half.

11. Inverse Trigonometric Functions

Sine is not one-to-one, so it has no inverse — until you cut its domain down until it is.

12. Sine Rule, Cosine Rule and Area

Triangles that are not right-angled — and the case where two different triangles fit the same data.

13. Heights and Distances

Angles of elevation and depression, bearings, and the diagram that does most of the work.

14. Harmonic Form

a sin θ + b cos θ is a single wave. That fact is what superposition means.