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=e−iz=cos(z)−isin(z)
from which we have cos(z)=2+√3 ?!
Method 2
I added ilog(2−√3) both sides, to get:
ilog(2−√3)+z=ilog(4)
from which we have, following similar procedure, cos(z)=2−√3, 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+e−iz2 and sinz=eiz−e−iz2i
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