Lesson Explainer: Domain and Range of Trigonometric Functions Mathematics

In this explainer, we will learn how to determine the domain and range of a trigonometric function.

Let us begin by recalling the definitions of the domain and range of a function.

Theorem: Domain and Range of a Function

The domain of a function 𝑓(𝑥) is the set of all possible values 𝑥 such that function 𝑓(𝑥) is defined.

The range of a function 𝑓(𝑥) is the set of all possible values the function 𝑓(𝑥) can take, when 𝑥 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 a graph of a function, its domain is the part of the horizontal axis where the graph exists, and its range is the part of the vertical axis where the graph exists.

One of the important characteristics of the graph of a trigonometric function is that the pattern of the graph repeats indefinitely. When the behavior of function 𝑓(𝑥) repeats over an interval of length 𝐿, then we say that 𝑓(𝑥) is periodic with period 𝐿. In other words, a periodic function with period 𝐿 must satisfy 𝑓(𝑥)=𝑓(𝑥+𝐿), for any 𝑥 in the domain.

To study the domain and the range of a periodic function, we first need to understand the domain and the range of the function in an interval of length 𝐿, for instance [0,𝐿]. We can choose any interval of length 𝐿, but it will be convenient in most cases to consider the interval [0,𝐿]. Since the function is periodic with period 𝐿, the function will repeat the same behavior outside of this interval. Hence, the range of the periodic function will be the same as its range over [0,𝐿]. Furthermore, if the function is defined for all values in [0,𝐿], the function will be defined for any values outside this interval as well due to its periodicity. In this case, the domain of the periodic function is all real numbers, denoted or ],[.

In our first example, we will determine the domain and the range of a periodic function from its given graph.

Example 1: Determining the Domain and Range of a Function from the Graph

Consider the following graph of 𝑓(𝜃).

  1. What is the domain of 𝑓(𝜃)?
  2. What is the range of 𝑓(𝜃)?

Answer

Before we answer the questions about the domain and range of 𝑓(𝜃), we note that the graph of the function repeats indefinitely. This means that the function 𝑓(𝜃) is a periodic function. We can see that the function has a local minimum at 𝜃=0 and returns to the same place at 𝜃=2𝜋. From there, the function repeats the same values. This means that the period of this function is 2𝜋.

Part 1

In this part, we need to determine the domain of a periodic function from the given graph. We noted that the period of 𝑓(𝜃) is 2𝜋. Recall that, if a periodic function is defined within an interval whose length is equal to the period, the function is defined for all real numbers. From the given graph, we can see that 𝑓(𝜃) is defined for all values within the interval [0,2𝜋], whose length is equal to the period 2𝜋.

Hence, the domain of 𝑓(𝜃) is all real numbers, or ],[.

Part 2

In this part, we need to determine the range of a periodic function from the given graph. We noted that the period of 𝑓(𝜃) is 2𝜋. Recall that if the range of a periodic function is the same as the range of the function over an interval whose length is equal to the period, the function is defined for all real numbers. From the given graph, we can see that the minimum value for the function 𝑓(𝜃) over the interval [0,2𝜋] is 4, and its maximum value over this interval is 6. Since 𝑓(𝜃) takes all values between its maximum and minimum, the range of 𝑓(𝜃) over this interval is [4,6].

Hence, the range of 𝑓(𝜃) is [4,6].

In the previous example, we considered the domain and range of a periodic function from the given graph. We can use the same method to find the domain and range of sine and cosine functions. Recall that the angle of 2𝜋 radians measures a full revolution on the unit circle. This means that the values of trigonometric ratios, sine and cosine, on the unit circle would remain the same if we add 2𝜋 radians to any angle. This means that, for any angle of 𝜃 radians, sinsincoscos𝜃=(𝜃+2𝜋),𝜃=(𝜃+2𝜋).

This tells us that the sine and cosine functions are periodic with a period of 2𝜋 radians. Hence, we can find the domain and range of the sine and cosine functions by considering the graph of these functions over an interval of length 2𝜋. Consider the graphs of sin𝜃 and cos𝜃.

Both graphs above are over the interval [3𝜋,3𝜋], but we only need the graph over an interval of length 2𝜋. So, we can consider this graph over the interval [0,2𝜋] to find the domain and range of these functions. Since both functions are defined everywhere in the interval [0,2𝜋], we know that the domains of both the sine and cosine functions are all real numbers.

We can also note that the minimum value of both functions over the interval [0,2𝜋] is 1, and the maximum value is 1. Since both functions take all values between the maximum and minimum, the range of sine and cosine functions over this interval, and hence over their domains, is [1,1].

We summarize these results as follows.

Definition: Domain and Range of Sine and Cosine Functions

The domain of functions sin𝜃 and cos𝜃 is all real numbers, denoted either ],[ or .

The range of functions sin𝜃 and cos𝜃 is [1,1].

Above, we determined the domain and the range of sin𝜃 and cos𝜃 by using the graph and the periodicity of these trigonometric functions. In our next example, we will use the same method to determine the domain and the range of a periodic function.

Example 2: Finding Domain and Range of a Periodic Function from its Graph

The following graph shows the function 𝑓(𝜃). Assume the function has a period of 2𝜋.

  1. What is the domain of 𝑓(𝜃)?
  2. What is the range of 𝑓(𝜃)?

Answer

We know that all characteristics of a periodic function are contained in the graph of a function over an interval whose length is equal to the period. This function has a period of 2𝜋; hence we only need to consider its graph over the interval of length 2𝜋. In this example, the function 𝑓(𝜃) is graphed over an interval with a length larger than 2𝜋, so this should provide sufficient information to determine the domain and the range of 𝑓(𝜃).

Part 1

The domain of a function is a set of all possible input values. From the given graph, we can see that the function is well defined at any values of 𝜃. Thus, the domain of 𝑓(𝜃) is all real numbers, or ],[.

Part 2

The range of a function is a set of all possible function values. From the given graph, we observe that this function oscillates between 7 and 3, taking all values in between. The graph never goes below 7 or above 3 in the vertical axis. Hence, the range of 𝑓(𝜃) is [7,3].

When we are given algebraic expressions of trigonometric functions, we can use functional transformations to find the range of a function by graphing functions 𝑎𝜃+𝑏sin or 𝑎𝜃+𝑏cos for some constants 𝑎 and 𝑏. Let us consider only the range of functions of the type 𝑎𝜃+𝑏sin, since the range of the latter will be identical.

We begin with the function sin𝑥 that has the range [1,1]. Multiplying a function by a positive constant 𝑎 results in a vertical dilation (stretching or contracting) by the scale factor 𝑎, which changes the range of the function from [1,1] to [𝑎,𝑎]. Multiplying a function by a negative constant results in a reflection over the 𝑥-axis followed by a dilation by the scale factor |𝑎|. In this case, the reflection over the 𝑥-axis does not change the range of this function since it is symmetric with respect to the 𝑥-axis. So, the vertical dilation by the factor |𝑎| makes the range of the function 𝑎𝑥sin be [|𝑎|,|𝑎|]. We note that this expression for the range is true whether 𝑎>0 or 𝑎<0.

Next, we know that adding 𝑏 to the function results in a vertical shift (upward if 𝑏>0 and downward if 𝑏<0) by 𝑏 units. Since the range of 𝑎𝑥sin is [|𝑎|,|𝑎|], a vertical shift by 𝑏 units would change this range to [|𝑎|+𝑏,|𝑎|+𝑏].

For example, let us consider the range of 2𝑥+1sin using the following diagram.

In the diagram, the solid blue graph represents the function sin𝑥 that has the range [1,1]. Multiplying sin𝑥 by 2 changes the range from [1,1] to [2,2]. Two sided blue arrows in the diagram indicate that the original graph is stretched vertically by a factor of 2 to obtain the graph of 2𝑥sin, giving the dashed curve. As we noted earlier, we can see that the range of 2𝑥sin is [2,2]. Adding 1 to 2𝑥sin shifts the range up by 1, leading to the new range [1,3]. The vertical red arrows indicate that the graph of 2𝑥sin is shifted upward to obtain the graph of 2𝑥+1sin. We can note that the range of this red graph is [1,3], as expected.

In the next example, we will determine the range of a sine function from its algebraic expression.

Example 3: Finding the Range of a Given Sine Function

Find the range of the function 𝑓(𝜃)=87𝜃sin.

Answer

Recall that the sine function, sin𝜃, is periodic and its graph oscillates between 1 and 1. This tells us that the range of sin𝜃 is [1,1]. We can use this information to determine the range of the given function.

The range is the set of all possible function values, so we are looking for all possible values of the expression 87𝜃,𝜃.sinwhereisanyrealnumber

Since 𝜃 can take any value, multiplying 𝜃 by 7 does not change the set of values that can result from this expression. Since sin𝜃 can take any values in [1,1], we know that sin7𝜃 is also limited to the same range.

Multiplying 8 to the expression sin7𝜃 stretches the graph of the function vertically by a factor of 8. This transformation changes the range from [1,1] to [8,8].

Hence, the range of 𝑓(𝜃)=87𝜃sin is [8,8].

In our next example, we will determine the domain and the range of a cosine function using the same method.

Example 4: Finding Domain and Range of Trigonometric Functions

Consider the function 𝑓(𝑥)=4(7𝑥+𝜋)+5cos.

  1. What is the domain of 𝑓(𝑥)?
  2. What is the range of 𝑓(𝑥)?

Answer

Part 1

Let us find the domain of 𝑓(𝑥). The domain of a function is the set of all possible input values. We know that the domain of function cos𝜃 is all real numbers. This tells us that there is no restriction for input values for cosine. In the function 𝑓(𝑥), the expression 7𝑥+𝜋 is inside the cosine function. Since this function is well defined for any real number, the domain of 𝑓(𝑥) is all real numbers, or ],[.

Part 2

Let us consider the range of 𝑓(𝑥). The range is the set of all possible function values, so we need to determine the set of all possible values for the expression 4(7𝑥+𝜋)+5,𝑥.cosforanyrealnumber

We know that the range of 7𝑥+𝜋 is all real numbers, so this expression can take any value. Denoting 𝜃=7𝑥+𝜋, we need to find the set of possible values of the expression 4𝜃+5,𝜃.cosforanyrealnumber

Let us think through function transformations to obtain the range of this expression. We know that cos𝜃 has the range [1,1]. Multiplying the function by 4 results in stretching this range vertically by the factor of 4, leading to the range [4,4]. Adding 5 to this expression shifts the function up by 5, which gives us the range [1,9].

Alternatively, we can find this algebraically by performing the following computations: 1𝜃144𝜃414𝜃+59.coscoscos

This also leads to the range [1,9].

The range of 𝑓(𝑥) is [1,9].

In our next example, we will identify an unknown constant in a trigonometric function when we are given the range of the function.

Example 5: Finding the Range of a Trigonometric Function from Its Rule

The range of the function 𝑓(𝜃)=𝑎3𝜃cos is 54,54. Find the value of 𝑎 where 𝑎>0.

Answer

Recall that the range of cos𝜃 is [1,1] and the domain of cos𝜃 is all real numbers. Since both 3𝜃 and 𝜃 have the same range, cos3𝜃 has the same range as cos𝜃. This tells us that the set of all possible outcomes, that is, the range, of cos3𝜃 is [1,1].

Multiplying a function by a positive constant 𝑎 results in a vertical dilation by the factor 𝑎. Since the range of cos3𝜃 is [1,1], applying a vertical dilation to this range makes the range of 𝑎3𝜃cos be [𝑎,𝑎].

We have shown that the range of 𝑓(𝜃)=𝑎3𝜃cos is [𝑎,𝑎]. We are given that the range of this function is 54,54. This leads to 𝑎=54.

So far, we have considered examples dealing with the domain and the range of functions involving sines and cosines. Let us turn our focus to the tangent function. Recall that the tangent function is defined as the ratio of the sine and cosine functions tansincos𝜃=𝜃𝜃.

This means that the tangent function will have domain restrictions when the cosine function equals zero. Consider the graph tan𝜃 over the interval of [3𝜋,3𝜋] radians.

From the graph, we can see that the tangent function repeats over 𝜋 radians. Hence, the period of the tangent function is 𝜋 radians, which is different from the period of the sine and cosine functions. This means that we can find the domain and range of the tangent function by examining its graph over the interval [0,𝜋]. Over this interval, we see that the tangent function is not defined at 𝜃=𝜋2. Since this function has period 𝜋, this means that the tangent function is not defined at every 𝜋 starting from the point 𝜋2, as we can see in the graph. In other words, tan𝜃 is undefined when 𝜃=𝜋2+𝑛𝜋,𝑛.

To find the range of tan𝜃, we can also consider its graph over the interval [0,𝜋]. We can see that the graph of the function over this interval goes up to the positive infinity and down to the negative infinity. This means that the range of tan𝜃 is the set of all real numbers.

We summarize these results as follows.

Definition: Domain and Range of Tangent Function

The domain of tan𝜃, in radians, is all real numbers except for 𝜃=𝜋2+𝑛𝜋,𝑛.

The domain of tan𝜃, in degrees, is all real numbers except for 𝜃=90+180𝑛,𝑛.

The range of tan𝜃 is all real numbers, denoted either ],[ or .

In our final example, we will identify the input values where a tangent function is undefined.

Example 6: Finding the Values Where Tangent is Undefined

Find the values of 𝜃 in radians such that the function 𝑓(𝜃)=(3𝜃)tan is undefined.

Answer

Recall that the domain of the tangent function tan𝜃, in radians, excludes values of the form 𝜃=𝜋2+𝑛𝜋,𝑛.

The given function 𝑓(𝜃)=(3𝜃)tan includes the tangent function, so we need to find the values of 𝜃 such that this function is not defined when the input 3𝜃 of the tangent gives one of these values. In other words, 𝑓(𝜃) is not defined when 3𝜃=𝜋2+𝑛𝜋,𝑛.

Dividing both sides of the equation above by 3, the function 𝑓(𝜃)=(3𝜃)tan is undefined when 𝜃 is equal to 𝜋6+𝑛𝜋3,𝑛.

Let us finish by recapping a few important points from the explainer.

Key Points

  • If a function 𝑓(𝑥) is periodic with period 𝐿, then we can find the domain and range of this function by considering the graph of this function over the interval [0,𝐿].
  • The domain of functions sin𝜃 and cos𝜃 is all real numbers, denoted either ],[ or .
    The range of functions sin𝜃 and cos𝜃 is [1,1].
  • For any constants 𝑎 and 𝑏, the range of the function 𝑎𝑥+𝑏sin or 𝑎𝑥+𝑏cos is [|𝑎|+𝑏,|𝑎|+𝑏].
  • The domain of tan𝜃, in radians, is all real numbers except for 𝜃=𝜋2+𝑛𝜋,𝑛. The domain of tan𝜃, in degrees, is all real numbers except for 𝜃=90+180𝑛,𝑛. The range of tan𝜃 is all real numbers, denoted either ],[ or .

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