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The law of sines and cosines

In an arbitrary triangle, without a right angle, the law of sines is used when we know a side and the angle opposite it, and the law of cosines when we know two sides and the angle between them.

\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}

the law of sines

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

the law of cosines

Symbols

$a,\ b,\ c$	sides of the triangle
$A,\ B,\ C$	angles opposite the sides $a,b,c$	°

Example

Two rods, 3.0 m and 5.0 m, are joined with $60^\circ$ between them: $c=\sqrt{3^2+5^2-2\cdot3\cdot5\cos60^\circ}=\sqrt{19}\approx 4.36$ m.

The law of sines can give two possible angles (

v

and

180^\circ-v

) - check which one fits the figure.

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Part of Foundations of Mathematics: Trigonometry and geometry.