**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

## 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 =

=

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