How do you simplify sin(sin^-1(1/2)+cos^-1(3/5))sin(sin−1(12)+cos−1(35))?

1 Answer
Alan P.
Jun 18, 2016

sin(sin^(-1)(1/2)+cos^(-1)(3/5))=color(green)(1.3)sin(sin−1(12)+cos−1(35))=1.3

Explanation:

sin^(-1)sin−1 is by definition an angle in [0,pi)[0,π)
within this interval sin(sin^(-1)(theta))=thetasin(sin−1(θ))=θ
color(white)("XXX")rarr sin(sin^(-1)(1/2))=1/2XXX→sin(sin−1(12))=12

cos^(-1)cos−1 is by definition an angle in (-pi/2,pi/2](−π2,π2]
within this range a cos=y/rcos=yr of 3/535 implies a standard triangle with
color(white)("XXX")x/r=sin=4/5XXXxr=sin=45
So cos^(-1)(3/5)=sin^(-1)(4/5)cos−1(35)=sin−1(45)
and, again, since this is in QI,
color(white)("XXX")sin(cos^(-1)(3/5))=sin(sin^(-1)(4/5))=4/5XXXsin(cos−1(35))=sin(sin−1(45))=45

Therefore
sin(sin^(-1)(1/2)+cos^(-1)(3/5))sin(sin−1(12)+cos−1(35))
color(white)("XXX")=1/2+4/5XXX=12+45

color(white)("XXX")=13/10 (=1 3/10 = 1.3)XXX=1310(=1310=1.3)

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