Complex Modulus/Examples/Tangent of Angle + i

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Example of Complex Modulus

$\left\vert{\tan \theta + i}\right\vert = \left\vert{\sec \theta}\right\vert$

where:

$\theta \in \R$ is a real number
$\tan \theta$ denotes the tangent function
$\sec \theta$ denotes the secant function.


Proof

\(\displaystyle \left\vert{\tan \theta + i}\right\vert\) \(=\) \(\displaystyle \left\vert{\tan \theta + 1 i}\right\vert\)
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\tan^2 \theta + 1}\) Definition of Complex Modulus
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {\sec^2 \theta}\)
\(\displaystyle \) \(=\) \(\displaystyle \left\vert{\sec \theta}\right\vert\) Absolute Value Equals Square Root of Square

$\blacksquare$


Sources