Triangles and vectors

In Chapter 1 we solved right triangles. Most triangles, though, have no right angle. This chapter gives two laws that solve any triangle, the law of sines and the law of cosines, and formulas for the area of a triangle.

It then introduces vectors: arrows that have both a size and a direction, such as a velocity or a force. Trigonometry is the tool for adding them and finding the angles between them.

This chapter covers the following topics:

The law of sines

Solving triangles that are not right triangles when you know two angles and a side, and the ambiguous case of two sides and an angle opposite one of them.

The law of cosines

Solving triangles when you know two sides and the angle between them, or all three sides, and choosing which law to use.

Area of a triangle

The area from two sides and the angle between them, and Heron’s formula for the area from three sides.

Vectors

Quantities with size and direction, components, adding vectors and multiplying by numbers, length and direction, and applications to velocity and force.

The dot product

The dot product of two vectors, the angle between vectors, perpendicular vectors, projections, and work.

Practice questions

Practice problems on every section of this chapter, with full step-by-step solutions.