Cosecant of Complex Number

Theorem

Let $a$ and $b$ be real numbers.

Let $i$ be the imaginary unit.

Then:

$\csc \paren {a + b i} = \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}$

where:

$\csc$ denotes the complex cosecant function.

$\sin$ denotes the real sine function

$\cos$ denotes the real cosine function

$\sinh$ denotes the hyperbolic sine function

$\cosh$ denotes the hyperbolic cosine function

Proof

\(\ds \csc \paren {a + b i}\)

\(=\)

\(\ds \frac 1 {\sin \paren {a + b i} }\)

Definition of Complex Cosecant Function

\(\ds \)

\(=\)

\(\ds \dfrac 1 {\sin a \cosh b + i \cos a \sinh b}\)

Sine of Complex Number

\(\ds \)

\(=\)

\(\ds \dfrac {\sin a \cosh b - i \cos a \sinh b} {\paren {\sin a \cosh b - i \cos a \sinh b} \paren {\sin a \cosh b + i \cos a \sinh b} }\)

multiplying denominator and numerator by $\sin a \cosh b - i \cos a \sinh b$

\(\ds \)

\(=\)

\(\ds \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b - i^2 \cos^2 a \sinh^2 b}\)

Difference of Two Squares

\(\ds \)

\(=\)

\(\ds \dfrac {\sin a \cosh b - i \cos a \sinh b} {\sin^2 a \cosh^2 b + \cos^2 a \sinh^2 b}\)

Definition of Imaginary Unit

$\blacksquare$

Also see

• Sine of Complex Number
• Cosine of Complex Number
• Tangent of Complex Number
• Secant of Complex Number
• Cotangent of Complex Number