SOLUTION: evaluate exactly: sin(sin^-1(5/13)+cos^-1(3/5))

Question 134530: evaluate exactly:
sin(sin^-1(5/13)+cos^-1(3/5))
Answer by nycsharkman(136) (Show Source): You can put this solution on YOUR website!
Evaluate exactly:
sin(sin^-1(5/13)+cos^-1(3/5))
This is not a log question.
This question involves finding the exact value of an expression that has inverse trigonometric functions.
We seek the sum of the two angles, A = sin^-1(5/13) and B = cos^-1(3/5).
For sin^-1(5/13), we use sinA = 5/13 and for cos^-1(3/5) we use cosB = 3/5.
We now use the Pythagorean Identities to find sinA and cosB. We know that both sinA and cosB BOTH = positive fractions.
Let sqrt = square root for short.
sinA = sqrt{1 - cos^2A}
sinA = sqrt{1 - (5/13)^2}
sinA = sqrt{1 - 25/169}
sinA = sqrt{144/169}
sinA = 12/13
We just found the value of sinA.
===============================
We now need to do the same with the other trig function.
cosB = sqrt{1 - sin^2B}
cosB = sqrt{1 - (3/5)^2}
cosB = sqrt{1 - 9/25}
cosB = sqrt{16/25}
cosB = 4/5
We just found the value of cosB.
================================
As a result,(sin^-1(5/13)+ cos^-1(3/5)) = one of the angle sum identities.
(sin^-1(5/13)+cos^-1(3/5)) = sin(A + B) = sinA cosB + cosAsinB
sinA = 12/13
cosB = 4/5
cosA = 3/5
sinB = 5/13
We now plug and chug.
12/13 times 4/5 PLUS 3/5 times 5/13
12/13 times 4/5 = 48/65
3/5 times 5/13 = 15/65
We now have this:
48/65 + 15/65 = 63/65
Final answer is 63/65
Got it?
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