How to find the range of asinx+bcosx+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
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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