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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..