### Video Transcript

In this video, we will learn how to
determine the domain and range of trigonometric functions. We will begin by recalling the
definitions of domain and range of a function. The domain of a function π of π₯
is a set of all possible values of π₯ such that the expression π of π₯ is
defined. The range of a function π of π₯ is
a set of all possible values that the expression π of π₯ can take, where π₯ is any
number from the domain of the function. In particular, we can find the
domain and range of a function from its graph. Given the graph of a function, the
domain is the part of the horizontal axis where the graph exists and the range is
the part of the vertical axis where the graph exists.

Letβs begin by considering the
graph of π¦ equals sin π₯ for values of π₯ from negative 360 to 360 degrees. We can see that the function is
well defined for every π₯-value. This means that the domain of sin
π₯ are all real numbers. This can also be written as the set
of numbers on the open interval from negative β to β. We see that the graph oscillates
between negative one and one. The maximum value of the graph is
one and the minimum value is negative one. This means that the possible values
of sin π₯ are between these two values. And the range of this function is
the set of values on the closed interval from negative one to one. The same is true of the cosine
function. This, once again, has a domain of
all real numbers and a range from negative one to one inclusive.

We can summarize this as
follows. The domain of the functions sin π₯
and cos π₯ is all real numbers denoted as shown. Note that these are often written
as sin π and cos π, where the function would be π of π. The range of the functions sin π₯
and cos π₯ is the set of numbers on the closed interval negative one to one. Letβs now consider how we can find
the domain and range of any periodic function from its graph.

The following graph shows the
function π of π. Assume the function has a period of
two π. What is the domain of π of π? What is the range of π of π?

We know that all the
characteristics of a periodic function are contained over an interval of this
length. In this question, we are told the
period is equal to two π. Therefore, we only need to consider
the graph between zero and two π. The domain of any function is a set
of all possible input values. And we can see from the graph that
the function is well defined at all values of π. We can therefore conclude that the
domain of π of π is all real numbers written as the open interval from negative β
to β.

The range of any function is the
set of all output values. From the graph, we see that the
function oscillates and is continuous between negative seven and three. The maximum value of the graph is
three, and the minimum value is negative seven. We can therefore conclude that the
range of π of π is the set of values on the closed interval from negative seven to
three. The two answers to this question
are the open interval from negative β to β and the closed interval from negative
seven to three.

We will now consider how the
transformation of trigonometric functions affects the domain and range. We recall that the sine function
had domain and range as shown. Any transformation to this function
would not alter its domain. However, certain transformations
will affect the range of our function.

Letβs consider the function π of
π₯, which is equal to π sin π₯ plus π, where π and π are real constants. Multiplying a function by a
positive constant π results in a vertical dilation or stretch by the scale factor
π. This would change the range of a
function from the closed interval negative one, one to the closed interval negative
π, π. However, multiplying a function by
a negative constant results in a reflection over the π₯-axis and a dilation by the
scale factor of the absolute value of π. This means that the range of the
function π sin π₯ is equal to the closed interval from the negative absolute value
of π to the positive absolute value of π.

Next, we know that adding π to a
function results in a vertical shift upwards if π is greater than zero and
downwards if π is less than zero. We can therefore conclude that the
range of the function π sin π₯ plus π is the closed interval from the negative
absolute value of π plus π to the positive absolute value of π plus π. Letβs now consider how this works
in practice.

Consider the function π of π₯ is
equal to four cos of seven π₯ plus π plus five. What is the domain of π of π₯? What is the range of π of π₯?

We begin by recalling that the
domain of any function is the set of all possible input values and that the domain
of the cos of π is all real values. In this question, the expression
seven π₯ plus π is inside the cosine function. As this expression is well defined
for any real number, the domain of π of π₯ is all real numbers, which can be
written as the open interval from negative β to β. We know that the range is the set
of output values. Since the range of seven π₯ plus π
is all real numbers, this expression can take any real value. We will therefore let this equal π
so that we have four cos π plus five.

We know that cos of π has a range
on the closed interval from negative one to one. So we need to consider how the
transformations of this function to four cos π plus five affect the range. Firstly, we have multiplied the
function by four, which results in stretching the range vertically by the factor of
four. This gives us the closed interval
from negative four to four. Adding five to this expression
shifts the function up by five. Negative four plus five is equal to
one, and four plus five is equal to nine. This means that the range of π of
π₯ is the closed interval from one to nine.

We could have solved the second
part algebraically using our knowledge of inequalities. We know that cos of π is greater
than or equal to negative one and less than or equal to one. Multiplying through by four, we
have four cos π is greater than or equal to negative four and less than or equal to
four. Adding five to each term in the
inequality, we have four cos π plus five is greater than or equal to one and less
than or equal to nine. This corresponds to the closed
interval from one to nine. The domain of the function four cos
of seven π₯ plus π plus five is the open interval from negative β to β, and its
range is the closed interval from one to nine.

Before moving on to one final
example, letβs consider the domain and range of the tangent function. Unlike the sine and cosine
functions, the tangent function has domain restrictions. Considering the graph of tan π
over the interval from negative 360 degrees to 360 degrees or negative two π
radians to two π radians, we note that the graph is undefined at 90 degrees, 270
degrees, negative 90 degrees, and negative 270 degrees. Since the tangent function is
periodic, this behavior repeats indefinitely every 180 degrees. We can therefore conclude that π₯
is not defined and the graph has an asymptote at values of π₯ equal to 90 degrees
plus 180 degrees multiplied by π, where π is any integer.

The domain of tan π₯ can therefore
be written as shown. It is all real numbers except for
π₯ is equal to 90 degrees plus 180 degrees multiplied by π. This can also be written in radians
as π₯ is equal to π over two plus ππ. Once again, π is any integer
value. The range of the tangent function
is all real values, which can be written as the open interval from negative β to
β. In our final example, we will
identify the input values where a tangent function is undefined.

Find the values of π in radians
such that the function π of π equals tan of three π is undefined.

We begin by recalling that the
domain of the tangent function in radians excludes values of the form π is equal to
π over two plus ππ, where π is an integer. In this question, we are given the
function π of π is equal to the tan of three π, and we wish to find the values
where this is undefined. We can therefore let three π equal
π over two plus ππ, where π is an integer value. Dividing both sides of this
equation by three, we have π is equal to π over six plus ππ over three. Once again, this is for all integer
values of π. The tan of three π is therefore
undefined for all values of π equal to π over six plus ππ over three where π is
an integer.

We will now finish this video by
summarizing the key points. The domain of the functions sin π
and cos π is all real numbers. And the range of these functions is
the set of numbers on the closed interval from negative one to one. For any constants π and π, the
range of the functions π sin π plus π or π cos π plus π is the closed interval
from the negative absolute value of π plus π to the absolute value of π plus
π. The domain of the tan of π written
in radians is all real numbers except for π is equal to π over two plus ππ,
where π is an integer. We can also write this in degrees,
where π over two is equal to 90 degrees and π is 180 degrees. Finally, the range of the tangent
function, tan π, is all real numbers.