mathematics / trigonometry / six ratios

The Six Ratios as Lengths

Every one of the six functions is the length of a segment you can point at on the unit circle. That is where the names tangent, secant and the "co" prefix come from.

60° = 1.0472 rad = π/3
  • sin 0.866 √3/2
  • csc 1.155 2/√3
  • cos 0.500 1/2
  • sec 2.000 2
  • tan 1.732 √3
  • cot 0.577 1/√3

A solid swatch is the function, a dashed one its reciprocal. Teal is the sine family, amber the cosine family, purple the tangent family.

Sine is the vertical drop from the point to the x axis, cosine is the horizontal run.

The names in this subject look arbitrary. They are not. Every one of the six functions is the length of a specific line segment in one drawing, and the names describe what those segments do.

Start with the two already known. Put the radius at angle θ on the unit circle. Drop a vertical line from the point on the rim to the x axis. That segment has length sin θ. Now measure along the x axis from the origin to the foot of that vertical. That segment has length cos θ.

This is the half chord from page 1. The full chord joins two points on the rim; the half chord runs from the rim to the midpoint of the chord, and on the unit circle it is exactly the sine.

There is a detail about signs that the picture cannot show. A drawn length is always positive. The functions are not: sin θ is negative below the x axis and cos θ is negative to the left of the origin. Read the drawn segments as magnitudes and take the sign from which side of the origin the segment lies on. Page 5 is entirely about getting those signs right.

Reading the figure

The unit circle with the radius at angle θ. Two segments are highlighted: the vertical one from the point down to the x axis, and the horizontal one from the origin to the foot of that vertical. Each is labelled with its function name and its live value. Rotate the radius and watch the two lengths trade: as one grows the other shrinks, and neither ever exceeds 1.

Show
Pairs
Build it up

Everything
crowded on purpose: use the pairs above to read it
Frame
Off: the frame holds segments out to 2.6 units, so tangent leaves it above about 69 degrees.

Solid: the function itself. Dashed: its reciprocal. Each chip above carries its own colour and dash. Grey dashed: the two lines that touch the circle, at x = 1 and at y = 1. Grey solid: the line through the origin and the point, which cuts the circle at two places. It is a whole line rather than a ray, which is what lets tangent and secant keep working past 90 degrees.

1−11−160°cos = 0.500sin = 0.866

Black: the radius, always exactly 1. Amber solid: cos = 0.500. Teal solid: sin = 0.866.

1 unit = 89 px.

Towards 90 degrees Steps through 80, 85, 89, 89.9 and 89.99 degrees, turning the zoom on and switching to the tangent pair, so you can watch the segment run away while the angle barely moves.