Reckoning / Number / Trigonometry
Number · plane trigonometry
Trigonometry
Sine and cosine are coordinates: walk an angle round a circle of radius 1, and the point you reach is (cos θ, sin θ). Everything else, including the tangent, the laws of sines and cosines and every navigator’s traverse table, follows from that picture.
drag the point
Angle
Green: sin θ (height). Blue: cos θ (width). Red: tan θ, the length cut off on the tangent line at x = 1. As a navigator’s bearing, angles run clockwise from north instead; the panel gives both.
Exact values
At a few angles the values are exact surds, from the half-square (45°) and the half-equilateral triangle (30°, 60°); 15° and 75° follow by the addition formulas, 18°, 36° and 72° from the regular pentagon. Each entry here is checked against the computer’s sine when the site is built.
| θ | sin θ | cos θ | tan θ |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 15° | (√6−√2)/4 | (√6+√2)/4 | 2−√3 |
| 18° | (√5−1)/4 | √(10+2√5)/4 | √(25−10√5)/5 |
| 30° | 1/2 | √3/2 | √3/3 |
| 36° | √(10−2√5)/4 | (1+√5)/4 | √(5−2√5) |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 72° | √(10+2√5)/4 | (√5−1)/4 | √(5+2√5) |
| 75° | (√6+√2)/4 | (√6−√2)/4 | 2+√3 |
| 90° | 1 | 0 | ∞ |
Solve a triangle
Name the sides a, b, c and the opposite angles A, B, C. Any three facts that include a side fix the triangle, with one exception: two sides and an angle that is not between them (SSA) can give two different triangles, one, or none. That is the ambiguous case.
Given
The solver is the same code that the build checks: 2 000 random triangles are recovered from every kind of data, and the number of SSA solutions is compared with a ruler-and-compass construction.
Units of angle
| Unit | Symbol | Size | In radians |
|---|---|---|---|
| turn | 1 turn | 360° | 2π rad |
| degree | 1° | 1/360 turn | π/180 rad |
| minute of arc | 1′ | 1/60° | π/10 800 rad |
| second of arc | 1″ | 1/60′ | π/648 000 rad |
| radian (SI) | 1 rad | 180/π ° ≈ 57.295 78° | arc length = radius |
| gon (grade) | 1 gon | 0.9° | π/200 rad |
| point of the compass | 1 point | 11.25° | π/16 rad |
| NATO mil | 1 mil | 360/6400 ° | π/3200 rad |
The 32 points of the mariner’s compass, 11¼° apart:
Navigators worked with printed tables of logarithms of sines, and of the haversine, hav θ = sin²(θ/2), which keeps every quantity positive and avoids sign errors. The same haversine formula computes the great-circle distances on the sailing page. Courses were once given in points (“west by south”); today they are three-figure bearings, 000° to 359°.