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M01.5 · Trigonometry and the unit circle

A free, illustrated module of 4 lessons and one interactive in M01 Math Foundations: Sine, cosine and tangent as ratios; The unit circle and polar coordinates; Radians and sine waves; The law of cosines and the angle-addition formulas.

M01.5 L0 · Foundation 4 lessons · about 100 min Live · in review

Part of M01 Math Foundations in the Mathematics track.

How an angle becomes a number. These four lessons start from right triangles, whose side ratios depend only on their angle, put those ratios on a circle of radius 1 so that every angle has a sine and a cosine, measure angles by the distance walked round that circle so that the sine unrolls into a wave, and end with the law of cosines and the sine and cosine of a sum of angles.

What you need first. The coordinate plane, Pythagoras’ theorem and the distance formula (M01.2), and functions and their inverses (M01.3). The code uses Python’s math module, so importing a module (P01.5) helps.

Lessons

1. Sine, cosine and tangent as ratios M01.5.1 25 min

   Right triangles with the same angle are scaled copies, so the ratio of any two of their sides depends only on the angle. Name those ratios sine, cosine and tangent, find their exact values at 30°, 45° and 60°, and use them to find missing sides and angles.

2. The unit circle and polar coordinates M01.5.2 25 min

   Put a right triangle inside a circle of radius 1 and the cosine and sine become the coordinates of a point. That gives every angle a sine and a cosine, with signs, proves that their squares add to 1, and lets any point be named by a distance and a direction.

3. Radians and sine waves M01.5.3 25 min

   Measure an angle by the length of arc it cuts from the unit circle and a full turn becomes 2π. Unroll the circle and the sine becomes a wave that repeats every 2π; then stretch it, slow it and slide it with an amplitude, a period and a phase.

4. The law of cosines and the angle-addition formulas M01.5.4 25 min

   Two formulas from two pictures. Put one corner of a triangle at the origin and the distance formula gives the law of cosines; stand one right triangle on another and reading one corner two ways gives the sine and the cosine of a sum of two angles.

The interactive

M01.5-X1 · Unit-circle explorer. Drag a point round the unit circle, or press play or step. The angle is shown in degrees and radians, the cosine and the sine appear as the two sides of a right triangle, and the sine wave is traced beside or below the circle as the point turns, over two full turns. It lives in The unit circle and polar coordinates.