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Trigonometric Functions

Angles in Standard Position

An angle consists of two rays with a common vertex. It can be thought of as a rotation of one ray, called the initial side, onto the second ray, called the terminal side. If the rotation is counterclockwise, then the angle has a positive measure and if the rotation is clockwise, then the angle has a negative measure.

When an angle is drawn in the x-y plane with its vertex at the origin and its initial side on the positive x-axis, we say that the angle is in standard position. Figure 1 shows examples of angles in standard position.

Figure 1

The Unit Circle

Figure 2 below shows a circle with center at the origin and radius 1. We call this circle a unit circle. Its equation is ..

             

                              Figure 2

Terminal Points on the Unit Circle

Suppose an angle in standard position in the x-y plane has measure t radians. Then the terminal side of the angle intersects the unit circle in a point, P, with coordinates (a, b). The point, P, is called the terminal point determined by t. Explore the following interactive example to see the terminal point on the unit circle determined by an angle.

Note: The angle can be measured in either degrees or radians, but we will use radian measure unless otherwise specified.

Definition of the Sine and Cosine functions

The sine function, usually written , is defined as follows:  , where b is the second coordinate of the point on the unit circle determined by an angle of measure t.

Similarly we can define the cosine function, usually written as , as

,  where  is the first coordinate of the point on the unit circle determined by an angle of measure t.

Notice that the sine and cosine are functions of a real number , the radian measure of an angle. Also, given any real number , there is an angle with measure  radians. So, the domain of these two trigonometric functions is the set of all real numbers.  Since the maximum output value is 1 and the minimum output value is -1, the range is the interval [-1, 1].

Example:

Find .

Solution:

An angle of measure radians intersects the unit circle at the point (-1, 0).  So, .

The next interactive example allows you to see other values of the sine and cosine functions.

Graphs of the Sine and Cosine Functions

Since the unit circle has a circumference of , the sine and cosine functions repeat their values over every interval of length . That is,  for any value of . We say that these functions are periodic with a period of . To see how the graphs of the sine and cosine functions look over the domain interval , see the following interactive example.

Transformations of the Graphs of the Sine and Cosine

Vertical Shift

            The graph of  can be obtained by shifting the graph of

graph of   vertically  units. If   is positive, the graph is shifted  units upward and if  is negative, the graph is shifted  units downward. The number  is called the vertical shift.

Amplitude

            The graph of  is the same as the graph of  stretched vertically by a factor of , if.  If , then the graph is stretched and reflected across the horizontal axis. The number  is called the amplitude of the function. Note that the amplitude is always positive.

Phase Shift

The graph of  can be obtained by shifting the graph of

graph of   horizontally  units. If  is positive, the graph is shifted  units right and if  is negative, the graph is shifted  units left. The number  is called the phase shift.

Period

The graph of  can be obtained by horizontally stretching or shrinking the graph of  . A horizontal shrink by a factor of k occurs if k >1 and a horizontal shrink by a factor of k occurs if k < 1. Since the sine function has period ,  the function  completes one cycle as  varies from 0 to . That is, for .  Solving this inequality for , we get  . So this function completes one cycle as t varies from 0 toand has period .

Note: all of the above also can be applied to the cosine function.

Summary

The sine and cosine functions

                             

have amplitude

, period , phase shift  and vertical shift .

One complete cycle of the graph occurs on the interval .

Example

Find the amplitude, period, phase shift and vertical shift of the function

.

Graph one complete cycle.

Solution

Comparing our problem to the general form , we see that

a = 1, b = 3, k = 2, and . This tells us that the amplitude is 3, the period is , the phase shift is , and the vertical shift is 1. One complete cycle of the graph occurs on the interval  One complete cycle of the graph is shown in Figure 1.

            Figure 1

The interactive examples below allow you to see more graphs of  for different values of the constants a, b, c and k..