How do you find the exact value of cos (arcsin 3/5 - arccos 1/2)?

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Answer 1 · 2016-12-20T14:36:20

$(4+3sqrt 3)/10$ .

Let $a=arc sin (3/5) in Q_1 and b = arc cos (1/2) in Q_1$ .

Then, $sin a = 3/5 and cos a = sqrt(1-sin^2 a)=sqrt(1-9/25)=4/5$ .

Likewise,

$cos b = 1/2 and sin b =sqrt(1-cos^2 b) = sqrt(1-1/4) = sqrt 3/2.$

Now, the given expression is

cos( a - b ) = cos a cos b + sin a sin b

$=(4/5)(1/2)+(3/5)(sqrt 3/2)$

$=(4+3sqrt 3)/10$ .