+0  
 
0
1122
2
avatar

Can you please explain why the following identity is true? Where does it come from: pi/24 = cot^(-1)(2+sqrt(2)+sqrt(3)+sqrt(6)). Thanks for any help.

Guest Dec 23, 2015

Best Answer 

 #1
avatar+89876 
+15

cot-1( 2 + sqrt(2) + sqrt(3) + sqrt(6) )  = pi/24

 

cot [ pi/24]  = 2 + sqrt(3) + sqrt(2) + sqrt(6)

 

cot [pi/24]  =

                                                              

cot [(1/2)pi/12]] =

 

1/ tan [(1/2)pi/12)] =

 

1/ [sqrt ( 1 - cos(pi/12))/ (1 + cos(pi/12)]  =  

 

sqrt( 1 + cos(pi/12)) / sqrt( 1 - cos(pi/12) ) =  [multiply top/bottom  by sqrt( 1 + cos(pi/12)) ]

 

[1 + cos (pi/12)] / sqrt( 1 - cos^2(pi/12)) =

 

[1 + cos(pi/12)] / sqrt( sin^2(pi/12)) =

 

[1 + cos(pi/12) ] / sin(pi/12)  =

 

1/sin(pi/12) + cos(pi/12) / sin(pi/12)  =

 

 

csc[pi/12] + cot[pi/12] =

 

1/ sin [ pi/12] + 1 / [tan [pi/12]  =

 

1 / [ sin[ pi/4 - pi/6]] + 1 / [ tan[ pi/4 - pi/6] ]

 

1/ [ sin(pi/4)cos(pi/6) - sin(pi/6)cos(pi/4)]  +  1/ ( [tan(pi/4] - tan[pi/6]] / [1 + tan(pi/4)tan(pi/6)] )  =

 

1 / [ sqrt(2)sqrt(3)/4] - (sqrt(2)/4) ] + [ 1 + 1/sqrt(3)]/ [1 - 1/sqrt(3)]  =

 

4 / [ sqrt(6) - sqrt(2)]  +  [ (sqrt(3) + 1)/ sqrt(3))]   / ( [ sqrt(3) - 1]/ sqrt(3) )

 

4 [ sqrt(6) + sqrt(2)]/ 4  +  [ sqrt(3) + 1] / [sqrt(3) - 1]  =  [ multiply the second term by [sqrt(3) + 1] on top/bottom ]

 

sqrt(6) + sqrt(2)  + [sqrt(3) + 1]^2 / 2  =

 

sqrt(6) + sqrt(2) + [3 + 2sqrt(3) + 1] / 2  =

 

sqrt(6) + sqrt(2) + [ 4 + 2 sqrt(3)] / 2  =

 

sqrt(6) + sqrt(2) + 2 + sqrt(3)  =

 

2 + sqrt(3) + sqrt(2) + sqrt(6)    ..........and the left side  =  the right side

 

 

 

cool cool cool

CPhill  Dec 23, 2015
edited by CPhill  Dec 23, 2015
edited by CPhill  Dec 23, 2015
edited by CPhill  Dec 23, 2015
 #1
avatar+89876 
+15
Best Answer

cot-1( 2 + sqrt(2) + sqrt(3) + sqrt(6) )  = pi/24

 

cot [ pi/24]  = 2 + sqrt(3) + sqrt(2) + sqrt(6)

 

cot [pi/24]  =

                                                              

cot [(1/2)pi/12]] =

 

1/ tan [(1/2)pi/12)] =

 

1/ [sqrt ( 1 - cos(pi/12))/ (1 + cos(pi/12)]  =  

 

sqrt( 1 + cos(pi/12)) / sqrt( 1 - cos(pi/12) ) =  [multiply top/bottom  by sqrt( 1 + cos(pi/12)) ]

 

[1 + cos (pi/12)] / sqrt( 1 - cos^2(pi/12)) =

 

[1 + cos(pi/12)] / sqrt( sin^2(pi/12)) =

 

[1 + cos(pi/12) ] / sin(pi/12)  =

 

1/sin(pi/12) + cos(pi/12) / sin(pi/12)  =

 

 

csc[pi/12] + cot[pi/12] =

 

1/ sin [ pi/12] + 1 / [tan [pi/12]  =

 

1 / [ sin[ pi/4 - pi/6]] + 1 / [ tan[ pi/4 - pi/6] ]

 

1/ [ sin(pi/4)cos(pi/6) - sin(pi/6)cos(pi/4)]  +  1/ ( [tan(pi/4] - tan[pi/6]] / [1 + tan(pi/4)tan(pi/6)] )  =

 

1 / [ sqrt(2)sqrt(3)/4] - (sqrt(2)/4) ] + [ 1 + 1/sqrt(3)]/ [1 - 1/sqrt(3)]  =

 

4 / [ sqrt(6) - sqrt(2)]  +  [ (sqrt(3) + 1)/ sqrt(3))]   / ( [ sqrt(3) - 1]/ sqrt(3) )

 

4 [ sqrt(6) + sqrt(2)]/ 4  +  [ sqrt(3) + 1] / [sqrt(3) - 1]  =  [ multiply the second term by [sqrt(3) + 1] on top/bottom ]

 

sqrt(6) + sqrt(2)  + [sqrt(3) + 1]^2 / 2  =

 

sqrt(6) + sqrt(2) + [3 + 2sqrt(3) + 1] / 2  =

 

sqrt(6) + sqrt(2) + [ 4 + 2 sqrt(3)] / 2  =

 

sqrt(6) + sqrt(2) + 2 + sqrt(3)  =

 

2 + sqrt(3) + sqrt(2) + sqrt(6)    ..........and the left side  =  the right side

 

 

 

cool cool cool

CPhill  Dec 23, 2015
edited by CPhill  Dec 23, 2015
edited by CPhill  Dec 23, 2015
edited by CPhill  Dec 23, 2015
 #2
avatar
0

Thank you CPhill. Brilliant!.

Guest Dec 23, 2015

27 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.