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
\(\ds \left\vert{\tan \theta + i}\right\vert\) | \(=\) | \(\ds \left\vert{\tan \theta + 1 i}\right\vert\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {\tan^2 \theta + 1}\) | 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
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 1.2$. The Algebraic Theory: Examples