Complex Modulus/Examples/Tangent of Angle + i

From ProofWiki
Jump to navigation Jump to search

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

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

$\blacksquare$


Sources