# Find Range of Sine Functions

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
$[ -2 , 2 ]$

#### Matched Problem 2:

Find the range of function $$f$$ defined by
$$f(x) = -\dfrac{1}{5}\sin (\dfrac{x}{\pi} + \pi)$$

### Example 3

Find the range of function $$f$$ defined by
$$f(x) = 0.1\sin (\dfrac{x}{\pi} + \pi) - 2$$

#### Solution to Example 3

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
$[ -2.1 \; , \; 1.9 ]$

#### Matched Problem 3:

Find the range of function $$f$$ defined by
$$f(x) = - 4\sin (\dfrac{x}{\pi} + \pi) + 3$$

## Answers to the Above Matched Problems

1) $$[-1 , 1]$$
2) $$[- \dfrac{1}{5} , \dfrac{1}{5}]$$
3) $$[-1 , 7]$$