SOLUTION: how do you find the exact value of the expression cos(sin^-1 1/3

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Question 958763: how do you find the exact value of the expression
cos(sin^-1 1/3 - tan^-1 1/2)
Found 2 solutions by lwsshak3, ikleyn:
Answer by lwsshak3(11628)   (Show Source):
You can put this solution on YOUR website!
how do you find the exact value of the expression
cos(sin^-1 1/3 - tan^-1 1/2)
sinx=1/3
cosx=√(1-sin^2(x))=√1-1/9=√(8/9)=√8/3
..
tany=1/2
hypotenuse of reference right triangle in quadrant I=√1+2^2=√5
cosy=2/√5
siny=1/√5
..
cos(sin^-1 1/3 - tan^-1 1/2)=cos(x+y)=cosx*cosy-sinx*siny
=√8/3*2/√5-1/3*1/√5=2√8/3√5-1/3√5=2√8/3√5-1/3√5=(2√8-1)/3√5
..
Check: w/calculator
sinx=1/3
x≈19.471˚
tany=1/2
y≈46.036
x+y=46.036˚
cos(sin^-1 1/3 - tan^-1 1/2)=cos(x+y)≈cos(46.036˚)≈0.6942
exact value=(2√8-1)/3√5≈0.6942

Answer by ikleyn(53937)   (Show Source):
You can put this solution on YOUR website!
.
how do you find the exact value of the expression
cos(sin^-1 1/3 - tan^-1 1/2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

        The solution in the post by @lwsshar3 is incorrect due to arithmetic error.
        I came to bring a correct solution.

So, they want you find cos(arcsin(1/3) - arctan(1/2)). An average school student will be shocked, I think. Therefore, let's move forward accurately. Let x = arcsin(1/3) and y = arctan(1/2). So, sin(x) = 1/3 and x is in QI; tan(y) = 1/2 and y is in QI, too. Then cos(x) = = = = = . For 'y', tan(y) = 1/2 is the same as to say sin(y) = , cos(y) = . Therefore, cos(arcsin(1/3)-acrtan(1/2)) = cos(x-y) = cos(x)*cos(y) + sin(x)*sin(y) = = + = + = = = . ANSWER. cos(arcsin(1/3) - arctan(1/2)) = .

Solved correctly.

SOLUTION: how do you find the exact value of the expression cos(sin^-1 1/3 - tan^-1 1/2)