Question 134530
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.

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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.

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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?