Domain and range of sine and cosine curves

The domain of a function is the set of all values that the independent variable (usually $x$x) can take and the range of a function is the set of all values that the dependent variable (usually $y$y) takes.  Each number in the range is the image of one or more domain values under the action of the function.

The domain and the range of continuous functions are usually defined using notations like

meaning $x$x is an element of the real numbers, or

$y\ge0$y≥0,

meaning $y$y is greater than or equal to zero. Combinations of these and similar notations are also used.

 

When specifying domains and ranges, a key idea is to identify values for which the domain does not exist. For the functions given by  $y=\sin x$y=sinx and $y=\cos x$y=cosx we see that there are no values in the set of real numbers that the variable $x$x cannot take. 

Since there are no values to exclude, the domain for both sine and cosine functions is the whole set of real numbers.

 

We may, if we wish, restrict the domain in any way we choose. Often we may only be interested in the behaviour of the function on an interval, rather than on the whole set of real numbers, for example, $0^\circ\le x\le360^\circ$0°≤x≤360°.

The range is the set of values that the function $y$y can take as $x$x varies over the domain.  We can see from the graphs that the $y$y-values vary continuously between the maximum and the minimum values. For both of these functions, the maximum is $1$1 and the minimum $-1$−1. Thus, the range is $-1\le y\le1$−1≤y≤1.

For other cosine and sine functions we would need to establish the maximum and minimum by either reading off the graph or by using the following rule:

The range of $y=a\sin\left(bx-c\right)+d$y=asin(bx−c)+d or $y=a\cos\left(bx-c\right)+d$y=acos(bx−c)+d is 

$d-a\le y\le d+a$d−a≤y≤d+a.

This formula uses facts about amplitude and vertical translation. The amplitude is the amount by which the function goes above the central value given by $y=d$y=d.

 

Example

Give the natural domain and the range of the function defined by $y(x)=2.5\cos x+0.5$y(x)=2.5cosx+0.5.

The function makes sense for all real numbers $x$x. That is, the natural domain is the set of real numbers.

When $\cos x=1$cosx=1, which is its maximum, $y(x)=3$y(x)=3. And when $\cos x=-1$cosx=−1, its minimum, $y(x)=-2$y(x)=−2.  So, the range of $y(x)$y(x) is $-2\le y\le3$−2≤y≤3.

The function varies by $2.5$2.5 above and below its mid-value of $0.5$0.5.

 

Worked Examples

Question 1

Question 2

Question 3