SOLUTION: Question Part Points Submissions Used 1 0/1 1/5 Total 0/1 Find the exact value of the expression in terms of x with the help of a reference triangle. sin(cos−1(3

Algebra ->  Trigonometry-basics -> SOLUTION: Question Part Points Submissions Used 1 0/1 1/5 Total 0/1 Find the exact value of the expression in terms of x with the help of a reference triangle. sin(cos−1(3      Log On


   

Question 1128737: Question Part
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Submissions Used
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1/5
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0/1

Find the exact value of the expression in terms of x with the help of a reference triangle.
sin(cos−1(3 − x))
Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
In the expression "sin(cos-1(3 - x))"
the "cos-1" or %22cos%22%5E%28-1%29 part refers to the inverse cosine function,
so cos%5E%28-1%29%283-x%29 is an angle A , between 0 and pi
(or between 0%5Eo and 180%5Eo if you want to visualize angles in degrees)
whose cosine is 3-x .
Those angles would have a non-negative sine, so sin%28cos%5E%28-1%29%283-x%29%29%3E=0 .
Using the trigonometric identity sin%5E2%28A%29%2Bcos%5E2%28A%29=1 ,
sin%5E2%28cos%5E%28-1%29%283-x%29%29%2Bcos%5E2%28A%29=1 ,
sin%5E2%28cos%5E%28-1%29%283-x%29%29=1-%283-x%29%5E2 ,
sin%5E2%28cos%5E%28-1%29%283-x%29%29=1-%289-6x%2Bx%5E2%29 ,
sin%5E2%28cos%5E%28-1%29%283-x%29%29=1-9%2B6x-x%5E2 , and
sin%5E2%28cos%5E%28-1%29%283-x%29%29=-8%2B6x-x%5E2
We are looking for a non-negative result, so
sin%28cos%5E%28-1%29%283-x%29%29=highlight%28sqrt%28-8%2B6x-x%5E2%29%29

NOTE:
Of course cos%5E%28-1%29%283-x%29 is not defined as a real number for all values of x .
It must be -1%3C=3-x%3C=1 or 2%3C=x%3C=4 .
That is the domain of the function,
but that is probably not expected as part of the answer.