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..




	Home / Review Topics 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..