Find the range of sine functions; examples and matched problems with their answers at the bottom of the page.

Graphical Analysis of Range of Sine Functions

The range of a function \( y = f(x) \) is the set of values \( y \) takes for all values of \( x \) within the domain of \( f \).
What is the range of \( y = f(x) = \sin(x) \)?
The domain of \( f \) above is the set of all values of \( x \) in the interval \( (-\infty, +\infty) \).
As \( x \) takes values from \( -\infty \) to \( +\infty \), \( \sin(x) \) takes all values between -1 and 1 as shown in the unit circle below. Hence

\( -1 \leq \sin(x) \leq 1 \) or \( -1 \leq y \leq 1 \)

In general, the range of any sine function of the form \( y = \sin (bx + c) \) is given by

\( -1 \leq \sin(bx + c) \leq 1 \) or \( -1 \leq y \leq 1 \)

Examples with Solutions

Example 1:

Find the range of function \( f \)
defined by

\( f(x) = -\sin (x) \)

Solution to Example 1

Start with the range of the basic sine function (see discussion above) and write
\( -1 \leq \sin(x) \leq 1 \)
Multiply all terms of the above inequality by -1 and change symbols of inequality to obtain
\( 1 \geq \sin(x) \geq -1 \)
which may also be written as
\( -1 \leq -\sin(x) \leq 1 \)
Hence the range of \( -\sin(x) \) is also given by the interval
\[ [ -1 , 1 ] \]

Matched Problem 1:

Find the range of
function \( f \) defined by

\( f(x) = -\sin (2 x) \)

Example 2

Find the range of function \( f \)
defined by

\( f(x) = 2\sin (-3 x - \dfrac{\pi}{6}) \)

Solution to Example 2

The range of \( \sin (-3 x - \dfrac{\pi}{6}) \) is given by
\( -1 \leq \sin (-3 x - \dfrac{\pi}{6}) \leq 1 \)
Multiply all terms of the above inequality by 2 to obtain the inequality
\( -2 \leq 2\sin (-3 x - \dfrac{\pi}{6}) \leq 2 \)
The range of the given function \( f \) is written above in inequality form and may also be written in interval form as follows
\[ [ -2 , 2 ] \]

The range of \( \sin (\dfrac{x}{\pi} + \pi) \) is given by
\( -1 \leq \sin (\dfrac{x}{\pi} + \pi) \leq 1 \)
Multiply all terms of the inequality by 0.1 to obtain
\( -0.1 \leq 0.1\sin (\dfrac{x}{\pi} + \pi) \leq 0.1 \)
Add -2 to all terms of the above inequality to obtain
\( -2.1 \leq 0.1\sin (\dfrac{x}{\pi} + \pi) -2 \leq 1.9 \)
The range of values of \( 0.1\sin (\dfrac{x}{\pi} + \pi
) -2 \) may also be written in interval form as follows
\[ [ -2.1 \; , \; 1.9 ] \]