What is sin(i) -- (sin of the imaginary number, i)
What is cos(i)-- cosine of the imaginary number, i
orrr
tan(1+i)
You can derive these things from the equation:
Exp(i x) = cos(x) + i sin(x) --->
sin(x) = [Exp(ix) - Exp(-ix)]/(2i)
cos(x) = [Exp(ix) + Exp(-ix)]/2
tan(x) =
1/i * [Exp(ix) - Exp(-ix)]/[Exp(ix) + Exp(-ix)]
These relations are also valid for complex values for x. E.g.:
tan(1+i) =
1/i * [Exp(i-1) - Exp(-i+1)]/[Exp(i-1) + Exp(-i+1)]
Exp[i-1] = Exp[i]Exp[-1] = Exp[-1](cos(1)+isin(1))
Exp[-i+1] = Exp[-i]Exp[1] = Exp[1](cos(1)-isin(1))
To evaluate sin(i), we can use the formula sin(x) = [Exp(ix) - Exp(-ix)]/(2i):
sin(i) = [Exp(i*i) - Exp(-i*i)]/(2i)
Since i*i = -1, we have:
sin(i) = [Exp(-1) - Exp(1)]/(2i)
Similarly, to evaluate cos(i), we can use the formula cos(x) = [Exp(ix) + Exp(-ix)]/2:
cos(i) = [Exp(i*i) + Exp(-i*i)]/2
Again, i*i = -1, so we get:
cos(i) = [Exp(-1) + Exp(1)]/2
Finally, to evaluate tan(1+i), we can use the tan(x) formula:
tan(x) = (1/i) * [Exp(ix) - Exp(-ix)]/[Exp(ix) + Exp(-ix)]
For tan(1+i), we substitute x = 1+i:
tan(1+i) = (1/i) * [Exp((1+i)i) - Exp(-((1+i)i))]/[Exp((1+i)i) + Exp(-((1+i)i))]
Now we can substitute Exp((1+i)i) and Exp(-((1+i)i)) using the exponential property and simplify the expression to get the result.