October 1, 2023

# How to find the range of asinx+bcosx+c

How to find the range of asinx+bcosx+c

Since and

is defined for all real values  of   , so the domain of the given  function is  the set of all real numbers. We have to find the range of  asinx+bcosx+c  . For the time being, assume that the quantity is not zero ( if it was zero, it would then mean that both a and b are zero, resulting in f(x)=asinx+bcosx+c being a constant function of value c . In that case, the range would have been just c.

Range of asinx+bcosx

# Maximum and minimum value of y=asinx+bcosx

To find the max. and min. value of  asinx+bcosx , we will use the identity    we have  . So we would  like to find an angle such that   and , for then we could write

Since  and  must be between 1 and 1, and  and  may not be in that range. Moreover, we know that  must equal 1, so we  scale everything by .

Let   and . Clearly  , so there is a unique angle such that    and and . Then

so .
Since , this implies
.

## Range of asinx+bcosx+c

Let , then .

We know that for all real  values of x

Hence the range of the function is

### Examples

Example1. Find the range of  cosx-sinx.

Here a=-1, b=1, c=0

Hence the range of cosx-sinx=

=

Example 2.  Find the range of -3sinx-4cosx -7

Sol.  Here a= -3, b=-4 and c=-7

So range of -3sinx-4cosx-7 =

= 