Let \(a, b, c\) be the sides opposite to the interior angles \(\alpha, \beta, \gamma\) respectively in a triangle. These formulas relate the sides and angles and are fundamental in triangle geometry and trigonometry.
\[ \frac{a}{\sin \alpha} = \frac{b}{\sin \beta} = \frac{c}{\sin \gamma} \]
Law of Sines: The ratio of a side to the sine of its opposite angle is constant for all sides of a triangle.
\[ c^2 = a^2 + b^2 - 2ab \cos \gamma \]
Law of Cosines: Relates the length of one side of a triangle to the lengths of the other two sides and the cosine of the included angle.
\[ \frac{a + b}{a - b} = \tan \frac{\alpha + \beta}{2} \tan \frac{\alpha - \beta}{2} \]
Side Ratio Identity: Relates sums and differences of sides with tangents of half-angle sums and differences.
\[ \frac{a + b}{c} = \cos \frac{\alpha + \beta}{2} \sec \frac{\gamma}{2} \]
Ratio involving sum of sides and half-angles: Expresses the ratio in terms of cosine and secant of half-angles.
\[ \frac{a - b}{c} = \sin \frac{\alpha - \beta}{2} \csc \frac{\gamma}{2} \]
Ratio involving difference of sides and half-angles: Expresses the ratio in terms of sine and cosecant of half-angles.
\[ \sin \alpha = \frac{2bc}{a + b + c} \sqrt{(s - a)(s - b)(s - c)} \]
Sine in terms of sides and semiperimeter: Provides a formula for \(\sin \alpha\) using side lengths and semiperimeter \(s\).
\[ s = \frac{1}{2}(a + b + c) \]
Semiperimeter: Half the perimeter of the triangle, used in many triangle area and relation formulas.