SOLUTION: Find the exact value of sin(cos^-1(12/13)+sin^-1(3/5))

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Question 737717: Find the exact value of sin(cos^-1(12/13)+sin^-1(3/5))
Found 2 solutions by lwsshak3, mananth:
Answer by lwsshak3(11628) About Me  (Show Source):
You can put this solution on YOUR website!
Find the exact value ofsin%28cos%5E-1%2812%2F13%29%2Bsin%5E-1%283%2F5%29%29
***
let O=opposite side
let A=adjacent side
let H=hypotenuse
...
let x=(cos^-1(12/13)
cos(x)=12/13=A/H
A=12, H=13
O=sqrt%28H%5E2-A%5E2%29=sqrt%2813%5E2-12%5E2%29=sqrt%28169-144%29=sqrt%2825%29=5
sin%28x%29=O%2FH=5%2F13
...
let y=sin^-1(3/5)
sin(y)=3/5=O/H
O=3, H=5
A=sqrt%28H%5E2-O%5E2%29=sqrt%285%5E2-3%5E2%29=sqrt%2825-9%29=sqrt%2816%29=4
cos%28y%29=A%2FH=4%2F5
..

...
Check with calculator:
cos(x)=12/13
x≈22.62º
sin(y)=3/5
y≈36.87º
22.62+36.87≈59.49º
sin(x+y)=sin(59.49º)≈0.8615..
As calculated, 56/65≈0.8615..

Answer by mananth(16946) About Me  (Show Source):
You can put this solution on YOUR website!
sin(sin^-1(5/13)+cos^-1(3/5))
let A = sin^-1(5/13) and B = cos^-1(3/5).
sin^-1(5/13), => sinA = 5/13 and cos^-1(3/5) => cosB = 3/5.
sinA and cosB are positive
sinA = sqrt{1 - cos^2A}
sinA = sqrt{1 - (5/13)^2}
sinA = sqrt{1 - 25/169}
sinA = sqrt{144/169}
sinA = 12/13

-----------
.
cosB = sqrt{1 - sin^2B}
cosB = sqrt{1 - (3/5)^2}
cosB = sqrt{1 - 9/25}
cosB = sqrt{16/25}
cosB = 4/5
---------------------
sin(A + B) = sinA cosB + cosAsinB
(sin^-1(5/13)+cos^-1(3/5)) =
sinA = 12/13
cosB = 4/5
cosA = 3/5
sinB = 5/13.
(12/13 )*( 4/5) +(3/5)* (5/13 )
12/13 * 4/5 = 48/65
3/5 *5/13 = 15/65
We get
48/65 + 15/65 = 63/65