Tangent of Angle minus Three Right Angles

Theorem

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

Proof

\(\ds \map \tan {x - \dfrac {3 \pi} 2}\)

\(=\)

\(\ds -\map \tan {x - \dfrac {\pi} 2}\)

as $\map \tan {x - \dfrac {\pi} 2}$ is in the opposite quadrant to $\map \tan {x - \dfrac {3 \pi} 2}$

\(\ds \)

\(=\)

\(\ds \cot \theta\)

Tangent of Complement equals Cotangent

$\blacksquare$

Also see

• Sine of Angle plus Three Right Angles
• Cosine of Angle plus Three Right Angles
• Cotangent of Angle plus Three Right Angles
• Secant of Angle plus Three Right Angles
• Cosecant of Angle plus Three Right Angles

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Angles larger than $90 \degrees$: Examples