Defining Sine and Cosine

Figure 7.1.6

Cos θ and sin θ are shown in Figure 7.1.6. Geometrically, if θ is between 0 and π/2 so that the point P(x, y) is in the first quadrant, then the radius OP is the hypotenuse of a right triangle with a vertical side sin θ and horizontal side cos θ. By Theorem 1, sin 9 and cos θ are real functions defined on the whole real line. We write sin" θ for (sin θ)", and cos" θ for (cos θ)". By definition (cos θ, sin θ) = (x, y) is a point on the unit circle x² + y² = 1, so we always have

sin² θ + cos² θ = 1.

Also,

- 1 ≤ sin θ ≤ 1, -1 ≤ cos θ ≤ 1.

Sin θ and cos θ are periodic functions with period 2π. That is,

sin (θ + 2πn) = sin θ, cos (θ + 2πn) = cos θ

for all integers n. The graphs of sin θ and cos θ are infinitely repeating waves which oscillate between - 1 and + 1 (Figure 7.1.7).

Figure 7.1.7

For infinite values of θ, the values of sin θ and cos θ continue to oscillate between -1 and 1. Thus the limits

lim_θ→∞ sin θ,     lim_θ→-∞ sin θ,

lim_θ→∞ cos θ,    lim_θ→-∞ cos θ,

do not exist. Figure 7.1.8 shows parts of the hyperreal graph of sin θ, for positive and negative infinite values of θ, through infinite telescopes.

Figure 7.1.8

The motion of our particle traveling around the unit circle with speed one starting at (1,0) (Figure 7.1.9) has the parametric equations

x = cos θ, y = sin θ.

Figure 7.1.9

The following table shows a few values of sin θ and cos θ, for θ in either radians or degrees.