GMY-04

Law of cosines

Any angle.

Reading speed

Geometrycosine law

Governing equation

c^2=a^2+b^2-2ab\\cos C

$C \in \left[1, 179\right]$

where

a: a (—)
b: b (—)
C: C (°)
c: c (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

00:0 / 00:08

①

Governing law — red

c^2=a^2+b^2-2ab\\cos C

$C \in \left[1, 179\right]$

$C = 1$ $c = 2.003$

Along the curve

②

Domain of the variable — blue

C_{\min} = 1,\quad C_{\max} = 179

$C \in \left[1, 179\right]$

$C = 1$ $C = 1$

Along the curve

③

Response curve — green

c^2=a^2+b^2-2ab\\cos C

$C \in \left[1, 179\right]$

$C = 1$ $c = 2.003$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}c}{\mathrm{d}C}

$C \in \left[1, 179\right]$

$C = 2.119$ $c^{\prime} = 0.011$

Along the curve

Lecture brief

Historical brief

Pythagoras, Heron, spherical trigonometry and the sphere’s volume are the measuring arts that predate calculus. The sheets compute length, area and angle on plane and sphere. This sheet (GMY-04 — Law of cosines) is the form associated with cosine law. Working symbols: $a$ , $b$ , $C$ $\rightarrow$ $c$ . Any angle.

Purpose

Purpose: compute

c

from

a

b

C

in Geometry via

c^2=a^2+b^2-2ab\\cos C

Any angle. Use it when a real geometry question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

a = 5.000\,\mathrm{—}

b = 7.000\,\mathrm{—}

C = 60.000\,\mathrm{^{\circ}}

, the governing relation

c^2=a^2+b^2-2ab\\cos C

yields

c = 6.2450\,\mathrm{—}

. Any angle. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

a a5 —b b7 —C C60 °

Outputs

c c6.2450 —

Reading speed

Watch on YouTube

Free library

Full library

Free PDF / open book

YouTube channels

00:0 / 00:08

Narration of this film

Any angle.

Reading speed

Watch on YouTube