Find the value of $csc( heta + i phi)$ on the unit circle.

To find the value of $\csc(\theta + i \phi)$ on the unit circle, we first recall that $\csc(z) = \frac{1}{\sin(z)}$ and we utilize the definition of the sine function for complex arguments.

Given $z = \theta + i \phi$, we have:

$\sin(z) = \sin(\theta + i \phi)$

Using the identity for sine of a complex number, we get:

$\sin(\theta + i \phi) = \sin(\theta) \cosh(\phi) + i \cos(\theta) \sinh(\phi)$

Therefore,

$\csc(\theta + i \phi) = \frac{1}{\sin(\theta + i \phi)} = \frac{1}{\sin(\theta) \cosh(\phi) + i \cos(\theta) \sinh(\phi)}$

Hence, the final value of $\csc(\theta + i \phi)$ is:

$\csc(\theta + i \phi) = \frac{\sin(\theta) \cosh(\phi) – i \cos(\theta) \sinh(\phi)}{\sin^2(\theta) \cosh^2(\phi) + \cos^2(\theta) \sinh^2(\phi)}$

To solve for $csc( heta + i phi)$ on the unit circle, we start by noting that $csc(z) = frac{1}{sin(z)}$ and use the complex sine function.

For $z = heta + i phi$, we have the identity:

$sin( heta + i phi) = sin( heta) cosh(phi) + i cos( heta) sinh(phi)$

Therefore, the cosecant function for the complex angle is:

$csc( heta + i phi) = frac{1}{sin( heta + i phi)} = frac{1}{sin( heta) cosh(phi) + i cos( heta) sinh(phi)}$

Using the conjugate, we get:

$csc( heta + i phi) = frac{sin( heta) cosh(phi) – i cos( heta) sinh(phi)}{sin^2( heta) cosh^2(phi) + cos^2( heta) sinh^2(phi)}$

To find $csc( heta + i phi)$ on the unit circle, we use the definition $csc(z) = frac{1}{sin(z)}$. For $z = heta + i phi$,

$sin( heta + i phi) = sin( heta) cosh(phi) + i cos( heta) sinh(phi)$

Thus,

$csc( heta + i phi) = frac{1}{sin( heta + i phi)} = frac{sin( heta) cosh(phi) – i cos( heta) sinh(phi)}{sin^2( heta) cosh^2(phi) + cos^2( heta) sinh^2(phi)}$

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