SOLUTION: Find the exact value of the expression.
sin(cos^-1 2/3 - tan^-1 1/4)
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Question 871093: Find the exact value of the expression.
sin(cos^-1 2/3 - tan^-1 1/4)
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
NOTE: I am rushing to get done before I have to go to my day job, so check for mistakes.
In this problem "cos^-1" means the inverse function of cosine,
and "tan^-1" means the inverse function of tangent.
We are looking for the exact value of
knowing that and ,
and that those inverse functions are defined so that
will be in quadrants I or II, and
will be in quadrants I or IV.
To calculate we can use the trigonometric identity
So we need to find , , and .
In quadrants I and I , so
Since only hapens in quadrants I and III,
and the definition of the inverse tangent function says that is in I or IV,
is definitely in quadrant I, where sine and cosine are positive.
This is how angle would look in a right triangle.
The hypotenuse of that right triangle measures
So, and
Substituting all the values found into ,
SInce that does not look elegant, and teachers do not like square roots in denominators, we multiply numerator and denominator times to get
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