Finding unknown quantities in triangles, such as the lengths of sides and the measure of angles, is a critical portion of the study of trigonometry. This Chapter deals with how to find such unknown quantities in different cases where certain quantities are known.
Also introduced in this Chapter are vectors. Consider for a moment the idea that all things cannot be represented with only a number. For example, while you can count the number of pencils in your desk with a number, you cannnot completely describe the force you apply when pushing on your desk with just a number. This is because, in addition to the number describing the strength of the force, you need something describing the direction of the force (in this case, pushing down). Vectors are a way to describe these kinds of quantities; they have both a magnitude and a direction.
This Chapter introduced ways to find unknown quantities in triangles when other quantities are known. Included were derivations and applications of the Law of Sines and the Law of Cosines. This was followed by alternate formulas for finding the area of a triangle, including Heron's Formula.
Cases where quantities were known, such as two angles and the not included side, or two angles and the included side, were presented, along with methods to find the other unknown quantities. This led to the "ambiguous case", where triangles with two sides and the not included angle as known quantities were addressed.
The Chapter then turned to vectors, including their definition, translation, addition, subtraction, multiplication by a scalar, and decomposition into combinations of unit vectors.