New Zealand
Level 8 - NCEA Level 3

# Domain and range of sine and cosine curves

Lesson

As we know from our work in other function areas, the domain of a graph is the set of all values that the independent variable (usually $x$x) can take and the range of a graph is the set of all values that the dependent variable (usually $y$y) can take.
For most functions we have covered already like linear, quadratic and cubic functions both the domain and range are usually defined using notation like

 Interpreted as $x$x is an element of the reals Read as $y$y, such that $y$y is greater than or equal to zero.

or a combination thereof.

When we were looking at how to define domains and ranges one of the key ideas was to first identify values for which the domain and range did not exist, and then move on from there.

Let's have a look at the generic sine and cosine curves $y=\sin x$y=sinx and $y=\cos x$y=cosx and define the domain and ranges for them.

The domain is the set of all values that $x$x can take,  We can see from the graph that there are no values to exclude, so the domain for both sine and cosine curves  is

.

The range is the set of values that y can take.  We can see from the graphs that the range of $y$y values exist between the maximum (distance of the amplitude above the central line) and the minimum (distance of the amplitude below the central line).  For both of these curves the maximum is $1$1 and the minimum $-1$1, thus the range is $-1\le y\le1$1y1. For other cosine and sine curves we would need to establish the maximum and minimum by either reading off the graph or using the equation.

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

$d-a\le y\le d+a$dayd+a

#### Worked Examples

##### Question 1

Consider the function $y=-3\sin x$y=3sinx, where $x$x is in radians.

1. State the domain of the function in interval notation.

2. State the range of the function in interval notation.

##### Question 2

Consider the function $y=-2\cos x$y=2cosx, where $x$x is in radians.

1. State the domain of the function in interval notation.

2. State the range of the function in interval notation.

##### Question 3

Consider the function $y=5\sin2x$y=5sin2x, where $x$x is in radians.

1. Graph the function on the axes below.

2. State the domain of the function in interval notation.

3. State the range of the function in interval notation.

##### Question 4

A sine function, $y$y, has the form $y=c\sin x$y=csinx and a range of $\left[-2,2\right]$[2,2]. Find an expression for $y$y.

### Outcomes

#### M8-2

Display and interpret the graphs of functions with the graphs of their inverse and/or reciprocal functions