Question 889393: My problem was (sin(π/3) cos(π/4)- sin(π/4)cos(π/3)²
I simplified that to (√6/4 - √2/4)² but I cant figure out what I should do next. Thank you for your help.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i believe this might be in the form of:
sin(a-b) = sin(a)cos(b) - cos(a) + sin(b), which is exactly the same as sin(a-b) = sin(a)cos(b) - sin(b)cos(a).
if so, then what you are really looking for is (sin(a-b)^2.
a-b would be equal to (pi/3 - pi/4) which would be equal to pi/12.
your problem then simplifies to (sin(pi/12))^2 which is equal to
.0669872981.
we should also be able to work it through directly as you tried to do.
pi/3 = 60 degrees.
pi/4 = 45 degrees.
sin(60) = sqrt(3)/2
cos(60) = 1/2
sin(45) = sqrt(2)/2
cos(45) = sqrt(2)/2
the problem looks like the following in degrees.
(sin(60)cos(45) - sin(45)cos(60))^2
i re-arranged sin(45) and cos(60) but that shouldn't matter because a*b = b*a.
replace sin(60) with sqrt(3)/2 and replace cos(45) with sqrt(2)/2 and replace sin(45) with sqrt(2)/2 and replace cos(60) with 1/2 to get the following:
(sqrt(3)/2 * sqrt(2)/2 - sqrt(2)/2 * 1/2))^2
since sqrt(3) * sqrt(2) = sqrt(6), this simplifies to:
(sqrt(6)/4 - sqrt(2)/4)^2
this looks like where you were at.
it does not appear that you can combine these, so you need to perform the operation indicated.
that operation is to square the expression.
squaring that expression uses the distributive property of multiplication.
(sqrt(6)/4 - sqrt(2)/4)^2 is equivalent to (sqrt(6)/4 - sqrt(2)/4) * (sqrt(6)/4 - sqrt(2)/4)
now it's easier to see how to apply the distributive property.
the distributive propoerty is performed as follows:
(a-b)*(a-b) = a^2 - ab - ba + b^2 when then simplifies to a^2 - 2ab + b^2.
you will get:
6/16 - sqrt(6)/4 * sqrt(2)/4 - sqrt(2)/4 * sqrt(6)/4 + 2/16
simplify this to get:
6/16 - (sqrt(6)*sqrt(2))/16 - (sqrt(2)*sqrt(6))/16 + 2/16
simplify further to get:
8/16 - sqrt(12)/16 - sqrt(12)/16
simplify further to get:
(8 - sqrt(12) - sqrt(12)) / 16
simplify further to get:
(8 - 2*sqrt(12))/16
simplify further to get:
(4 - sqrt(12))/8
i believe you can simplify one step further to get:
1/2 - sqrt(12)/8
i don't believe you can simplify further than that.
your solution, in decimal form, would be .0669872981.
this is exactly the answer we got above when we used the sin(a-b) formula which confirms that the expression could have been converted up front, although doing that would have meant a decimal format answer without the corresponding exact answer because you would have been dealing with sin(15) degrees which doesn't translate easily to exact format, if at all.
in this case, it does translate because we know that .0669872981 came from the exact answer of 1/2 - sqrt(12)/8. only problem is, if you started with .0669872981, how would you know that it came from 1/2 - sqrt(12)/8. you wouldn't.
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