Projection Formula⚠ unverified
Trigonometry / Triangle Solving · Side from the projection formula
Parameters
| Input | Symbol | Unit | Default | Description |
|---|---|---|---|---|
| a | a | m | 3 | Length |
| b | b | m | 4 | Length |
| A | A | degree | 30 | Angle |
| B | B | degree | 60 | Angle |
| Output | Symbol | Unit | Description |
|---|---|---|---|
| c | c | m | Third side |
The science & history
Understanding the Parameters
- a (m) — Length.
- b (m) — Length.
- A (degree) — Angle.
- B (degree) — Angle.
- Output c (m) — Third side.
How to Calculate
- Enter Length as
a(default 3 m). Use the unit menu when you need a different unit. - Enter Length as
b(default 4 m). Use the unit menu when you need a different unit. - Enter Angle as
A(default 30 degree). Use the unit menu when you need a different unit. - Enter Angle as
B(default 60 degree). Use the unit menu when you need a different unit. - Click Calculate. The card evaluates $c=a\cos B+b\cos A$ and shows the result in the declared output unit; Result in (when present) converts that number.
The Science: A Rigorous Derivation (Approaching a Proof)
Any triangle is determined (up to the SSA ambiguous case) by three independent pieces of $\{a,b,c,A,B,C\}$. The law of sines $a/\sin A = 2R$ is the extended law of sines: each side subtends the same circumcircle. The law of cosines $c^2 = a^2+b^2-2ab\cos C$ is Pythagoras with a correction for the projection of $a$ onto $b$; it specialises to Pythagoras at $C=90^\circ$ and rearranges to solve for an angle when three sides are known. The circumradius $R = abc/(4\Delta)$ and inradius $r = \Delta/s$ connect area to the two canonical circles.
Dimensional check. The declared output unit is m; ToolBox evaluates the
formula in SI (radians internally for every trigonometric call) and python-calc converts to
the unit on the card.
History and Development
The plane sine law is in al-Tūsī and, in the West, in Regiomontanus (De triangulis, 1464). The cosine law is Euclid II.12–13 in geometric language; Viète writes it trigonometrically. The law of tangents (Thomas Fincke, 1583) was a logarithm-friendly computational device for ASA/AAS in the era of Napier's bones.
Related Concepts: Law Of Sines Side, Law Of Sines Angle, Law Of Cosines Side, Law Of Cosines Angle, Law Of Tangents, Circumradius, Inradius