Laws of sines and cosines
Sine, cosine and tangent ratios need a right angle. How do you solve a triangle that has none?
Where you are 6 inside
What this area is about
Most triangles have no right angle. Two laws extend trigonometry to every triangle ABC, with side a across from angle A, and so on. Which law applies depends on which three parts you know, and once enough parts are known, the area follows too.
How these fit together
The two laws. The law of sines, \displaystyle \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}, solves a triangle from two angles and a side (AAS or ASA). The law of cosines, c^2 = a^2 + b^2 - 2ab\cos C, handles two sides and the included angle (SAS) or three sides (SSS), and contains the Pythagorean theorem as the case C = 90^\circ.
A warning case. The ambiguous case: two sides and an angle not between them (SSA) can give zero, one or two triangles. For an acute angle A the height h = b\sin A decides the count (for a right or obtuse A, whether a > b), and because \sin\theta = \sin(180^\circ - \theta), an angle found by inverse sine may have an obtuse partner.
Area. Area from two sides and the included angle, \displaystyle K = \frac{1}{2}ab\sin C, and Heron’s formula, from three sides alone.
Use. Applications of oblique triangles chooses among all of these for surveying and navigation.
A path through it
The law of sines first: the ambiguous case, the sine area formula and the law of cosines all build on it. Heron’s formula comes after the law of cosines, and the applications come last. Next, the law of cosines leads to the dot product, and the sine area formula to the cross product as area.