Question

find the value of cotz=-3i

Answer 1

cot z= -3i what is the value of z?

Some relations we can use.

cos(z) = (1/2)(e^(iz) + e^(-iz)),     sin(z) = (1/2)(e^(iz) – e^(-iz))

Let u = e^(iz) in the expression cot(z) = -3i, then

cot(z) = cos(z)/sin(z) = (e^(iz) + e^(-iz)) / (e^(iz) – e^(-iz)) = (u + u^(-1)) / (u – u^(-1))

(u^2 + 1) / (u^2– 1) = -3i

u^2 + 1= -3i.(u^2 – 1)

u^2(1 + 3i) = -1 + 3i

u^2 = -(1 - 3i) / (1 + 3i) = -(1 – 3i)^2 / {(1 + 3i)(1 – 3i) = -(1 – 3i)^2 / ( 1 + 9)

u^2 = -(1 – 6i – 9) / 10

u^2 = (8 + 6i)/10

u = √((4 + 3i)/5)

Then, using e^(iz) = u,

e^(iz) = √((4 + 3i)/5)

iz = ln{√((4 + 3i)/5)}

z = -i. ln{√((4 + 3i)/5)}