**Introduction to Primary Circular Functions**

Consider a circle in the Cartesian *xy* -plane with the following equation:

*x* ^{2} + *y* ^{2} = 1

This equation, as defined earlier in this chapter, represents the unit circle. Let *θ* be an angle whose apex is at the origin, and that is measured counterclockwise from the *x* axis, as shown in Fig. 12-1. Suppose this angle corresponds to a ray that intersects the unit circle at some point *P* = ( *x* _{0} , *y* _{0} ). We can define the three basic circular functions, also called the *primary circular functions* , of the angle *θ* in an elegant way. But before we get into this, let’s extend our notion of angles to cover negative values, and also to cover values more than 360° (2π rad).

**Offbeat Angles**

In trigonometry, any *direction angle* expressible as a real number, no matter how extreme, can always be reduced to something that is at least 0° (0 rad) but less than 360° (2π rad). If you look at Fig. 12-1, you should be able to envision how this works. Even if the ray *OP* makes more than one complete revolution counterclockwise from the *x* axis, or if it turns clockwise instead, its direction can always be defined by some counterclockwise angle of least 0° (0 rad) but less than 360° (2π rad) relative to the *x* axis.

**Fig. 12-1.** The unit circle is the basis for the trigonometric functions.

Any offbeat direction angle such as 730° or –9π/4 rad can be reduced to a direction angle that measures at least 0° (0 rad) but less than 360° (2π rad) by adding or subtracting some whole-number multiple of 360° (2π rad). But you have to be careful about this. A direction angle specifies orientation only. The orientation of the ray *OP* is the same for an angle of 540° (3π rad) as for an angle of 180° (π rad), but the larger value carries with it the insinuation that the ray (also called a *vector* ) *OP* has revolved 1.5 times around, while the smaller angle implies that it has undergone less than one complete revolution. Sometimes this doesn’t matter, but often it does!

Negative angles are encountered in trigonometry, especially in graphs of functions. Multiple revolutions of objects are important in physics and engineering. So if you ever hear or read about an angle such as –90°, –π/2 rad, 900°, or 5π rad, you can be confident that it has meaning. The negative value indicates clockwise rotation. An angle that is said to measure more than 360° (2π rad) indicates more than one complete revolution counterclockwise. An angle that is said to measure less than –360° (–2π rad) indicates more than one revolution clockwise.

**The Sine Function**

**Fig. 12-1.** The unit circle is the basis for the trigonometric functions.

In Fig. 12-1, imagine ray *OP* pointing along the *x* axis, and then starting to rotate counterclockwise on its end point *O* , as if point *O* is a mechanical bearing. The point *P* , represented by coordinates ( *x* _{0} , *y* _{0} ), thus revolves around point *O* , following the perimeter of the unit circle.

Imagine what happens to the value of *y* _{0} (the ordinate of point *P* ) during one complete revolution of ray *OP* . The ordinate of *P* starts out at *y* _{0} = 0, then increases until it reaches *y* _{0} = 1 after *P* has gone 90° or π/2 rad around the circle ( *θ* = 90° = π/2). After that, *y* _{0} begins to decrease, getting back to *y* _{0} = 0 when *P* has gone 180° or π rad around the circle ( *θ* = 180° = π). As *P* continues on its counterclockwise trek, *y* _{0} keeps decreasing until, at *θ* = 270° = 3π/2, the value of *y* _{0} reaches its minimum of –1. After that, the value of *y* _{0} rises again until, when *P* has gone completely around the circle, it returns to *y* _{0} = 0 for *θ* = 360° = 2π.

The value of *y* _{0} is defined as the *sine* of the angle *θ* . The *sine function* is abbreviated sin, so we can state this simple equation:

sin *θ* = *y* _{0}

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