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Trigonometric Functions

Domain and Range of Trigonometric Functions

Before we get into the domain and range of trigonometric functions, let’s understand what is a domain and range of any function. A function is nothing but a rule which is applied to the values inputted. The set of values that can be used as inputs for the function is called the domain of the function.

For e.g. for the function f(x) = √x, the input value cannot be a negative number since the square root of a negative number is not a real number. A range of a function is the set of output values for different input values. For e.g. for the function f(x) = x2 + 5, the range would be {5, 6, 7, …..}.

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Domain and Range of Trigonometric Functions

We know that the sine and cosine functions are defined for all real numbers. We also know that for each real number ‘x’,

-1 ≤ \( \sin{x} \) ≤ 1 and -1 ≤ \( \cos{x} \) ≤ 1.

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Therefore,

  • the domain of y = \( \sin{x} \) and y = \( \cos{x} \) is the set of all real numbers
  • the range is the interval [-1, 1], or -1 ≤ y ≤ 1.

Cosec x or \( \csc{x} \)

We know that, \( \csc{x} \) = \( \frac{1}{\sin {x}} \). Therefore,

  • the domain of y = \( \csc{x} \) is the set {x: x ∈ R and x ≠ nπ, n ∈ Z}
  • the range is the set {y: y ∈ R, y ≥ 1 or y ≤ -1}

\( \sec{x} \)

We know that, \( \sec{x} \) = \( \frac{1}{\cos {x}} \). Therefore,

  • the domain of y = \( \sec{x} \) is the set {x: x ∈ R and x ≠ (2n + 1)\( \frac{\pi}{2} \), n ∈ Z }
  • the range is the set {y: y ∈ R and y ≤ -1 or y ≥ 1}

\( \tan{x} \)

We know that, \( \tan{x} \) = \( \frac{\sin {x}}{\cos{x}} \). Therefore,

  • the domain of y = \( \tan{x} \) is the set {x: x ∈ R and x ≠ (2n + 1)\( \frac{\pi}{2} \), n ∈ Z }
  • the range is the set of all real numbers

\( \cot{x} \)

We know that, \( \cot{x} \) = \( \frac{\cos {x}}{\sin{x}} \). Therefore,

  • the domain of y = \( \cot{x} \) is the set {x: x ∈ R and x ≠ nπ, n ∈ Z}
  • the range is the set of all real numbers.

The following table describes the behavior of these trigonometric functions in all four quadrants where x increases from 0 to \( \frac{\pi}{2} \), \( \frac{\pi}{2} \) to π, π to \( \frac{3\pi}{2} \), and \( \frac{3\pi}{2} \) to 2π.

Quadrant I Quadrant II Quadrant III Quadrant IV
sin increases from 0 → 1 decreases from 1 → 0 decreases from 0 → -1 increases from -1 → 0
cos decreases from 1 → 0 decreases from 0 → -1 increases from -1 → 0 increases from 0 → 1
tan increases from 0 → ∞ increases from -∞ → 0 increases from 0 → ∞ increases from -∞ → 0
cot decreases from ∞ → 0 decreases from 0 → -∞ decreases from ∞ → 0 decreases from 0 → -∞
sec increases from 1 → ∞ increases from -∞ → -1 decreases from -1 → -∞ decreases from ∞ → 1
cosec decreases from ∞ → 1 increases from 1 → ∞ increases from -∞ → -1 decreases from -1 → -∞

Graphical Representations of Trigonometric Functions

We already know that the values of \( \sin{x} \) and \( \cos{x} \) repeat after an interval of 2π. This can be shown as follows:

domain and range

domain and range

Hence, the values of \( \sec{x} \) and \( \csc{x} \) will also repeat after an interval of 2π. This can be shown as follows:

domain and range

domain and range

However, the values of \( \tan{x} \) repeat after an interval of π. Also, the values of \( \cot{x} \) which is the inverse of \( \tan{x} \) will repeat after an interval of π. This can be shown as follows:

domain and range

domain and range

Let’s look at some examples on domain and range of trigonometric functions now:

Example 1

If \( \cos{x} \) = – \( \frac{3}{5} \), where x lies in the third quadrant, then find the values of other five trigonometric functions.

Solution: Since \( \cos{x} \) = – \( \frac{3}{5} \), we have \( \sec{x} \) = – \( \frac{5}{3} \).

Now, we know that, \( \sin^2{x} \) + \( \cos^2{x} \) = 1
∴ \( \sin^2{x} \) = 1 – \( \cos^2{x} \) = 1 – \( \frac{9}{25} \) = \( \frac{16}{25} \)
∴ \( \sin{x} \) = ± \( \frac{4}{5} \).

However, according to the problem, x lies in the third quadrant. Hence,
\( \sin{x} \) = – \( \frac{4}{5} \)

Since, \( \csc{x} \) = \( \frac{1}{\sin {x}} \), we have
\( \csc{x} \) = – \( \frac{5}{4} \)

Finally, \( \tan{x} \) = \( \frac{\sin {x}}{\cos {x}} \). Therefore,
\( \tan{x} \) = \( \frac{-\frac{4}{5}}{-\frac {3}{5}} \) = \( \frac{4}{3} \).

And, \( \cot{x} \) = \( \frac{1}{\tan {x}} \). Therefore,
\( \cot{x} \) = \( \frac{3}{4} \).

Learn Compound Angles here in detail. 

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Range of Trigonometric Functions

More Solved Examples for You

Question 1: If \( \cot{x} \) = \( \frac{3}{4} \), where x lies in the third quadrant, then find the values of other five trigonometric functions.

Answer : We know that, \( \cot{x} \) = \( \frac{1}{\tan {x}} \). Therefore,
\( \tan{x} \) = \( \frac{1}{\cot {x}} \) = \( \frac{1}{\frac {3}{4}} \) = \( \frac{4}{3} \).

We also know that,
\( \sec^2{x} \) = 1 + \( \tan^2{x} \) = 1 + \( (\frac{4}{3})^2 \) = 1 + \( \frac{16}{9} \) = \( \frac{25}{9} \)
∴ \( \sec{x} \) = ± \( \frac{5}{3} \).

Since x lies in the third quadrant, the value of \( \sec{x} \) will be negative. Therefore,
\( \sec{x} \) = – \( \frac{5}{3} \).
Next, \( \cos{x} \) = \( \frac{1}{\sec {x}} \) = \( \frac{1}{-\frac {5}{3}} \) = – \( \frac{3}{5} \)

Now, \( \tan{x} \) = \( \frac{\sin {x}}{\cos {x}} \). Hence,
\( \sin{x} \) = \( \tan{x} \).\( \cos{x} \) = (\( \frac{4}{3} \)) x (\( -\frac{3}{5} \)) = – \( \frac{4}{5} \).
Finally, \( \csc{x} \) = \( \frac{1}{\sin {x}} \) = \( \frac{1}{-\frac {4}{5}} \) = – \( \frac{5}{4} \).

Question 2: What is meant by domain and range?

Answer: Domain means the set of possible input values. The domain of a graph involves all the input values which are represented on the x-axis. The range means the set of possible output values whose representation takes place on the y-axis.

Question 3: How can one find the domain of a function?

Answer: One can determine the domain of each function by looking for independent variables values which one is allowed to use. Usually one must avoid 0 on the fraction’s bottom and the negative values which are under the square root sign.

Question 4: How can one find the range of a function?

Answer: One can find the range of a function by the following steps:

  • The range of a function happens to the spread of possible y-values.
  • Substitution of different x-values into the expression for y so as to understand what is going on.
  • One must look for the minimum as well as the maximum values of y.
  • Represent this by drawing a sketch

Question 5: What is meant by a function?

Answer: A function refers to an expression that shows the relationship between the one variable and another variable.

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