Law Of Sines

{a \over \sin A}={b \over \sin B}={c \over \sin C}=2R\,

where ''R'' the is radius of the triangle's Circumcircle . This formula is useful to compute the remaining sides of a triangle if two angles and a side are known, a common problem in the technique of Triangulation . It can also be used when two sides and one of the non-enclosed angles are known; in this case, the formula may give two possible values for the enclosed angle. When this happens, often only one result will cause all angles to be less than 180°; in other cases, there are two valid solutions to the triangle.

It can be shown that:
:

2R = rac{abc} {2\sqrt{s(s-a)(s-b)(s-c)}} = rac {2abc} {\sqrt{(a^2+b^2+c^2)^2-2(a^4+b^4+c^4) }}

where ''s'' is the semi-perimeter,

:

s = rac{(a+b+c)} {2}.

DERIVATION

Make a triangle with sides ''a'', ''b'', and ''c'', and opposite angles ''A'', ''B'', and ''C''. Make a line from the angle ''C'' to its opposite side ''c'' that cuts the figure into two right triangles, and call the length of this line ''h''.

It can be observed that:
:

\sin A = rac{h}{b}

and

\; \sin B = rac{h}{a}.

Therefore:
:

h = b\,(\sin A) = a\,(\sin B)

and
:

rac{a}{\sin A} = rac{b}{\sin B}.

Doing the same thing with the line drawn between angle ''A'' and side ''a'' will yield:
:

rac{b}{\sin B} = rac{c}{\sin C}.

EXAMPLES

Here is an example of how to solve a problem using the law of sines:

Given: side ''a'' = 10, side ''c'' = 7, and angle ''C'' = 30 degrees

Using the law of sines, we know that :

rac{a}{\sin A} = rac{c}{\sin C}.

Plugging in the given values, we find that :

rac{10}{\sin A} = rac{7}{\sin 30}.

Simplifying, the sine of angle ''A'' is equal to 5/7, or approximately 0.714. Thus, angle ''A'' is equal to 45.58 degrees.

Or another example of how to solve a problem using the law of sines:

If two sides of the triangle are equal to ''R'' and the length of the third side, the Chord , is given as 100' (30.48 m) and the angle ''C'' ''opposite'' to the chord is given in '''''degrees''''', then angle ''A'' = angle ''B'' = :

{(180-C) \over 2}

and

:

{R \over \sin A}={\mbox{chord} \over \sin C}\,

''or''

{R \over \sin B}={\mbox{chord} \over \sin C}\,

{\mbox{chord} \,\sin A \over \sin C} = R

''or''

{\mbox{chord} \,\sin B \over \sin C} = R

This is North American Railroad Surveying practice.

SEE ALSO

Triangulation

Law Of Cosines

Law Of Tangents

Surveying

EXTERNAL LINKS

Excellent tutorial on the law of sines

The Law of Sines at Cut-the-knot

Degree of Curvature

Information About

Law Of Sines