Tuesday, 20 June 2017

trigonometry - Evaluate cos(z), given that z=ilog(2+sqrt3)




Q. Evaluate cos(z), given that z=ilog(2+3)



I did it in 2 ways, and the results do not match.



Method 1



Multiply both sides by i3, we get



iz=log(2+3) and so,




2+3=eiz=cos(z)isin(z)



from which we have cos(z)=2+3 ?!



Method 2



I added ilog(23) both sides, to get:



ilog(23)+z=ilog(4)




from which we have, following similar procedure, cos(z)=23, which fortunately is <1



What actually is happening? And is cos(z) even defined for complex number z?


Answer



Hint: For complex z, we simply have cosz=eiz+eiz2 and sinz=eizeiz2i


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