SOLUTION: write an algebraic expression for the expression: sec[arcsin(x-1)]

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Question 103305: write an algebraic expression for the expression:
sec[arcsin(x-1)]
Answer by Fombitz(13823) About Me  (Show Source):
You can put this solution on YOUR website!
Let's work this one backwards.
Imagine a triangle with a hypotenuse of 1.
drawing%28+300%2C+300%2C+-.2%2C+1.2%2C+-.2%2C1.2%2C%0D%0Agreen%28line%28++0%2C+0%2C.87%2C.6%29%29%2C%0D%0Agreen%28line%28.87%2C.6%2C.87%2C+0%29%29%2C%0D%0Agreen%28line%28++0%2C+0%2C.87%2C+0%29%29%2C%0D%0Alocate%28.9%2C.3%2Cx-1%29%2C%0D%0Alocate%28.4%2C.5%2C1%29%2C%0D%0Alocate%28.2%2C.1%2CA%29%29
From this diagram,
sin+%28A%29+=+%28x-1%29%2F1
or in other words
A=arcsin%28x-1%29
Then
sec%28A%29+=+sec%28arcsin%28x-1%29%29
sec%28A%29+=+1%2Fcos%28A%29
drawing%28+300%2C+300%2C+-.2%2C+1.2%2C+-.2%2C1.2%2C%0D%0Agreen%28line%28++0%2C+0%2C.87%2C.6%29%29%2C%0D%0Agreen%28line%28.87%2C.6%2C.87%2C+0%29%29%2C%0D%0Agreen%28line%28++0%2C+0%2C.87%2C+0%29%29%2C%0D%0Alocate%28.9%2C.3%2Cx-1%29%2C%0D%0Alocate%28.4%2C.5%2C1%29%2C%0D%0Alocate%28.6%2C-.05%2Cy%29%2C%0D%0Alocate%28.2%2C.1%2CA%29%29
Using the new variable y, you can say,
cos%28A%29+=+y%2F1
sec%28A%29+=+1%2Fy
From Pythagorean theorem,
%28x-1%29%5E2%2By%5E2=1
y%5E2=1-%28x-1%29%5E2
y=sqrt%281-%28x-1%29%5E2%29 To be technically correct, it can be + or - sqrt%281-%28x-1%29%5E2%29
sec%28A%29=1%2Fsqrt%281-%28x-1%29%5E2%29
sec%28arcsin%28x-1%29%29=%28%2B-1%29%2F%28sqrt%281-%28x-1%29%5E2%29%29