Tangent of Angle plus Three Right Angles

Theorem

$\map \tan {x + \dfrac {3 \pi} 2} = -\cot x$


Proof

\(\ds \map \tan {x + \frac {3 \pi} 2}\) \(=\) \(\ds \frac {\map \sin {x + \frac {3 \pi} 2} } {\map \cos {x + \frac {3 \pi} 2} }\) Tangent is Sine divided by Cosine
\(\ds \) \(=\) \(\ds \frac {-\cos x} {\sin x}\) Sine of Angle plus Three Right Angles and Cosine of Angle plus Three Right Angles
\(\ds \) \(=\) \(\ds -\cot x\) Cotangent is Cosine divided by Sine

$\blacksquare$


Also see


Sources

  • 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 5$: Trigonometric Functions: Functions of Angles in All Quadrants in terms of those in Quadrant I
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