SOLUTION: the answer to this problem tan(cos^-1(7/15)) as a fraction rationalizing the denominator if necessary

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Question 552548: the answer to this problem tan(cos^-1(7/15))
as a fraction rationalizing the denominator if necessary
Found 2 solutions by Alan3354, Theo:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
tan(cos^-1(7/15))
as a fraction
---------------
A = arccos(7/15)
sin%28A%29+=+sqrt%281+-+cos%5E2%287%2F15%29%29+=+sqrt%28176%2F225%29
tan = sin/cos = %28sqrt%28176%29%2F15%29%2F%287%2F15%29+=+sqrt%28176%29%2F7
tan = 4sqrt%2811%29%2F7

Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
problem is:
tan (cos^-1(7/15))
first fine cosine^-1(7/15)
that equals 62.181860723 degrees.
then find tan (62.181860723)
that equals 1.895214166
that's doing it using the calculator.
now to do it the way i think they want it.
you have a triangle.
call it ABC
assume it's a right triangle.
AB is the hypotenuse of this right triangle.
BC is the vertical leg of this right triangle.
AC is the horizontal leg of this right triangle.
the length of AC is 7.
the length of AB is 15
the length of BC is calculated to be sqrt(176) using the pythagorus theorem of a^2 + b^2 = c^2 where a and b are legs of a right triangle and c is the hypotenuse.
the angle you are working with is angle x which is the same as angle BAC.
cosine^-1 (7/15) points to angle x which is the same as angle BAC.
that's because cosine(x) = adj/hyp = 7/15.
tan(x) = opp/adj = sqrt(176)/7.
you have:
cosine^-1(7/15) = x
tan(x) = sqrt(176)/7
since sqrt(176)/7 equals 1.895214166, this method provides the exact same answer as the method i used up top.
a picture of your triangle is shown below:
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