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.
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?
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}
The Sine Wave
If you graph the position of the ball, as seen by your friend, with respect to time, the result is a sine wave (Fig. 1-7), which is a graphical plot of a sine function. Some sine waves are “taller” than others (corresponding to a longer string), some are “stretched out” (corresponding to a slower rate of rotation), and some are “squashed” (corresponding to a faster rate of rotation). But the characteristic shape of the wave is the same in every case. When the amplitude and the wavelength are multiplied and divided by the appropriate numbers (or constants), any sine wave can be made to fit exactly along the curve of any other sine wave.
Fig. 1-7. Position of orbiting ball as seen edge-on, as a function of time.
You can whirl the ball around faster or slower than one revolution per second. The string can be made longer or shorter. These adjustments alter the height and/or the frequency of the sine wave graphed in Fig. 1-7. But the fundamental rule always applies: the sine wave can be reduced to circular motion. Conversely, circular motion in the ( x,y ) plane can be defined in terms of a general formula:
y = a sin b θ
where a is a constant that depends on the radius of the circle, and b is a constant that depends on the revolution rate.
The Cosine Function
Look again at Fig. 1-5.
Fig. 1-5 . The unit circle is the basis for the trigonometric functions.
Imagine, once again, a ray from the origin outward through point P on the circle, pointing right along the x axis, and then rotating in a counterclockwise direction.
What happens to the value of x _{0} (the abscissa of point P ) during one complete revolution of the ray? The abscissa of P starts out at x _{0} = 1, then decreases until it reaches x _{0} = 0 when θ – 90° = π/2. After that, x _{0} continues to decrease, getting down to x _{0} = –1 when θ = 180° = π. As P continues counterclockwise around the circle, x _{0} begins to increase again; at θ = 270° = 3π/2, the value gets back up to x _{0} = 0. After that, x _{0} increases further until, when P has gone completely around the circle, it returns to x _{0} = 1 for θ = 360° = 2π.
The value of x _{0} is defined as the cosine of the angle θ. The cosine function is abbreviated cos. So we can write this:
cos θ = x _{0}
The Tangent Function
Once again, look at Fig. 1-5.
Fig. 1-5 . The unit circle is the basis for the trigonometric functions.
The tangent (abbreviated tan) of an angle θ is defined using the same ray OP and the same point P = ( x _{0} , y _{0} ) as is done with the sine and cosine functions. The definition is:
tan θ = y _{0} /x _{0}
Because we already know that sin θ = y _{0} and cos θ = x _{0} , we can express the tangent function in terms of the sine and the cosine:
tan θ = sin θ /cos θ
This function is interesting because, unlike the sine and cosine functions, it “blows up” at certain values of θ. Whenever x _{0} = 0, the denominator of either quotient above becomes zero. Division by zero is not defined, and that means the tangent function is not defined for any angle θ such that cos θ = 0. Such angles are all the odd multiples of 90° (π/2 rad).
Primary Circular Functions Practice Problems
Practice 1
What is tan 45°? Do not perform any calculations. You should be able to infer this without having to write down a single numeral.
Solution 1
Draw a diagram of a unit circle, such as the one in Fig. 1-5, and place ray OP such that it subtends an angle of 45° with respect to the x axis. That angle is the angle of which we want to find the tangent. Note that the ray OP also subtends an angle of 45° with respect to the y axis, because the x and y axes are perpendicular (they are oriented at 90° with respect to each other), and 45° is exactly half of 90°. Every point on the ray OP is equally distant from the x and y axes; this includes the point ( x _{0} , y _{0} ). It follows that x _{0} = y _{0} , and neither of them is equal to zero. From this, we can conclude that y _{0} / x _{0} = 1. According to the definition of the tangent function, therefore, tan 45° = 1.
Practice Problems for these concepts can be found at: The Circle Model Practice Test
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From Trigonometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.