SOLUTION: Find the exact values for sin2a, cos2a and tan2a given sin a= 1/2 and 0<a<pi/2. Thanks!!!

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Question 838594: Find the exact values for sin2a, cos2a and tan2a given sin a= 1/2 and 0 Thanks!!!
Answer by LinnW(1048) About Me  (Show Source):
You can put this solution on YOUR website!
We will use the equivalent expressions sin(2x) = 2 sin x cos x
Since sin(a) = 1/2 and sin(a) = (opposite side)/hypotenuse
To find the relative length of the third leg
1%5E2+%2B+t%5E2+=+2%5E2
1+%2B+t%5E2+=+4
t%5E2+=+3
t+=+sqrt%283%29
opp = 1 , adj = sqrt(3) , hyp = 2
Since cos(x) = adj/hyp ,
cos(a) = sqrt(3)/2
Substituting sqrt(3)/2 for cos(a) and 1/2 for sin(a) in 2sin(a)*cos(a)
2*(1/2)(sqrt(3)/2) = (sqrt(3)/2)
Now we have
sin(2a)= %28sqrt%283%29%2F2%29
-------------
cos(2a) = cos^2(a) - sin^2(a)
cos(2a) = %28sqrt%283%29%2F2%29%5E2+-+%281%2F2%29%5E2
cos(2a) = 3%2F4+-+1%2F4
cos(2a) = 2%2F4
cos(2a) = 1%2F2
-------------
tan(2a) = sin(2a)/cos(2a)
= %28sqrt%283%29%2F2%29+%2F+%28+1%2F2+%29
= sqrt%283%29
Summarizing
sin(2a)= %28sqrt%283%29%2F2%29,cos(2a) = 1%2F2,tan(2a) = sqrt%283%29
You may have noticed that for sin(a) = 1/2 a is 30 degrees
sin(60)= %28sqrt%283%29%2F2%29,cos(60) = 1%2F2,tan(60) = sqrt%283%29