Unit Circle, Length of an Arc, Sector

In this chapter we shall study the trigonometric functions, i.e., the sine and cosine function and other functions that are built up from them. Let us start from the beginning and introduce the basic concepts of trigonometry.

The unit circle

x² + y² = 1

has radius 1 and center at the origin.

Two points P and Q on the unit circle determine an arc PQ, an angle ∠ POQ, and a sector POQ. The arc starts at P and goes counterclockwise to Q along the circle. The sector POQ is the region bounded by the arc PQ and the lines OP and OQ. As Figure 7.1.1 shows, the arcs PQ and QP are different.

Figure 7.1.1

Trigonometry is based on the notion of the length of an arc. Lengths of curves were introduced in Section 6.3. Although that section provides a useful background, this chapter can also be studied independently of Chapter 6. As a starting point we shall give a formula for the length of an arc in terms of the area of a sector. (This formula was proved as a theorem in Section 6.3 but can also be taken as the definition of arc length.)

This formula can be seen intuitively as follows. Consider a small arc PQ of length Δs (Figure 7.1.2). The sector POQ is a thin wedge which is almost a right triangle of altitude one and base Δs. Thus Δa ~ jΔs. Making Δs infinitesimal and adding up, we get A = js.

Figure 7.1.2 

The number π ~ 3.14159 is defined as the area of the unit circle. Thus the unit circle has circumference 2π.

The area of a sector POQ is a definite integral. For example, if P is the point P(l, 0) and the point Q(x, y) is in the first quadrant, then we see from Figure 7.1.3 that the area is

Notice that A(x) is a continuous function of x. The length of an arc has the following basic property.

 Figure 7.1.3