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 andmust be between and , and and may not be in that range. Moreover, we know that must equal , 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
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 =
Also read :