Trig Circle Help

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By — McGraw-Hill Professional
Updated on Oct 3, 2011

Introduction to Trig Circles and Primary Circular Functions

Consider a circle in rectangular coordinates 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. 1-5. Suppose this angle corresponds to a ray that intersects the unit circle at some point P = ( x 0 , y 0 ). We can define three basic trigonometric functions, called circular functions, of the angle θ in a simple and elegant way.

The Circle Model Primary Circular Functions

Fig. 1-5 . The unit circle is the basis for the trigonometric functions.

Circular Motion

Suppose you swing a glowing ball around and around at the end of a string, at a rate of one revolution per second. The ball describes a circle in space (Fig. 1-6A). Imagine that you make the ball orbit around your head so it is always at the same level above the ground or the floor; that is, so that it takes a path that lies in a horizontal plane. Suppose you do this in a dark gymnasium. If a friend stands several meters away, with his or her eyes right in the plane of the ball’s orbit, what will your friend see?

The Circle Model Primary Circular Functions Circular Motion

Fig. 1-6. Orbiting ball and string. At A, as seen from above; at B, as seen edge-on.

Close your eyes and use your imagination. You should be able to envision that the ball, seen from a few meters away, will appear to oscillate back and forth in a straight line (Fig. 1-6B). It is an illusion: the glowing dot seems to move toward the right, slow down, then stop and reverse its direction, going back toward the left. It moves faster and faster, then slower again, reaching its left-most point, at which it stops and turns around again. This goes on and on, at the rate of one complete cycle per second, because you are swinging the ball around at one revolution per second.

The Sine Function

The ray from the origin (point O ) passing outward through point P can be called ray OP. Imagine ray OP pointing right 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 ), therefore 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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