Question

How do you find cot 5pi/12 using identities

Answer 1

1/cos(5pi/12) In order to find the exact value of cos(5pi/12), you must express 5pi/12 as the sum or difference of two known unit circle values. Since 5pi/12 is not among our known values of the unit circle, we cannot solve this directly. This involves some experimentation. Can we split 5pi/12 as 4pi/12 + pi/12? 4pi/12 reduces to pi/3, and pi/3 is a known unit circle value. However, pi/12 is NOT a known unit circle value, so we try again. 5pi/12 = 3pi/12 + 2pi/12? This reduces to pi/4 + pi/6. Both are known unit circle values!! Now, we use the cosine addition property: cos(a+ b) = cos(a)cos(b) - sin(a)sin(b) Therefore, 1/cos(5pi/12) = 1/cos(3pi/12 + 2pi/12) = 1/cos(pi/4 + pi/6) = 1/[ cos(pi/4)cos(pi/6) - sin(pi/4)sin(pi/6) ] = 1/[ (sqrt(2)/2) (sqrt(3)/2) - (sqrt(2)/2) (1/2) ] = 1/[ sqrt(6)/4 - sqrt(2)/4 ] = 1/ ( [ sqrt(6) - sqrt(2)]/4 ) So we then just take the reciprocal. 4/[ sqrt(6) - sqrt(2) ] Of course, the simplified answer would probably have the denominator rationalized. Multiply top and bottom by the bottom's conjugate. 4(sqrt(6) + sqrt(2)) / [ 6 - 2 ] 4(sqrt(6) + sqrt(2)) / [ 4 ] Which reduces to just sqrt(6) + sqrt(2)"

Answer 2

Alternatively tan(x/2) = sin x/(1 + cos x) = (1 - cos x)/ sin x , as you say = (1 + sqrt(3)/2)/(1/2) = 2 + sqrt(3) hence cot(5pi/2) = 1/(2 + sqrt(3)) = (2 - sqrt(3)) / [ (2 + sqrt(3)) (2 - sqrt(3)) ] = (2 - sqrt(3)) / [ 2^2 - sqrt(3)^2 ] = (2 - sqrt(3)) / (4 - 3) = 2 - sqrt(3)