Spherical law of cosines

In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry.

Given a unit sphere, a "spherical triangle" on the surface of the sphere is defined by the great circles connecting three points $u, v$ , and $w$ on the sphere (shown at right). If the lengths of these three sides are $a$ (from $u$ to $v), b$ (from $u$ to $w$ ), and $c$ (from $v$ to $w$ ), and the angle of the corner opposite $c$ is $C$ , then the (first) spherical law of cosines states:

$\cos c=\cos a\cos b+\sin a\sin b\cos C\,$

Since this is a unit sphere, the lengths $a, b$ , and $c$ are simply equal to the angles (in radians) subtended by those sides from the center of the sphere. (For a non-unit sphere, the lengths are the subtended angles times the radius, and the formula still holds if $a, b$ and $c$ are reinterpreted as the subtended angles). As a special case, for $C = /* start https://en.wikipedia.org/ */ .mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px} /* end https://en.wikipedia.org/ */ ⁠π/2⁠$ , then $cos C = 0$ , and one obtains the spherical analogue of the Pythagorean theorem:

$\cos c=\cos a\cos b\,$

If the law of cosines is used to solve for $c$ , the necessity of inverting the cosine magnifies rounding errors when $c$ is small. In this case, the alternative formulation of the law of haversines is preferable.

A variation on the law of cosines, the second spherical law of cosines, (also called the cosine rule for angles) states:

$\cos C=-\cos A\cos B+\sin A\sin B\cos c\,$

where $A$ and $B$ are the angles of the corners opposite to sides $a$ and $b$ , respectively. It can be obtained from consideration of a spherical triangle dual to the given one.