Domain and Range of Trigonometric Functions

The domain of a function is the specific set of values that the independent variable in a function can take on. The range is the resulting values that the dependant variable can have as x varies throughout the domain.

Domain and range for sine and cosine functions

There are no restrictions on the domain of sine and cosine functions; therefore, their domain is such that x ∈ R. Notice, however, that the range for both y = sin(x) and y = cos(x) is between -1 and 1. Therefore, transformations of these functions in the form of shifts and stretches will affect the range but not the domain.

$graph of sin(x) and cos(x)$

The domain and range for tangent functions

Notice that y = tan(x) has vertical asymptotes at:

$\pm\frac{(2n+1)\pi}{2}$

Therefore, its domain is such that:

$x\neq\pm\frac{(2n+1)\pi}{2}$

However, its range is such at y ∈ R, because the function takes on all values of y. In this case, transformations will affect the domain but not the range.

$the graph of tangent function$

Example: Find the domain and range of y = cos(x) – 3

Solution:

Domain: x ∈ R

Range: - 4 ≤ y ≤ - 2, y ∈ R

Notice that the range is simply shifted down 3 units.

Example: Find the domain and range of y = 3 tan(x)

Solution:

Domain: x ∈ R

$x\neq\pm\frac{(2n+1)\pi}{2}$

Notice that the domain is the same as the domain for y = tan(x) because the graph was stretched vertically—which does not change where the vertical asymptotes occur.

Range: y ∈ R

Example 1:

Example 2: