Tangent of Three Right Angles less Angle

Theorem

$\tan \left({\dfrac {3 \pi} 2 - \theta}\right) = \cot \theta$

where $\tan$ and $\cot$ are tangent and cotangent respectively.

Proof

\(\ds \tan \left({\frac {3 \pi} 2 - \theta}\right)\)

\(=\)

\(\ds \frac {\sin \left({\frac {3 \pi} 2 - \theta}\right)} {\cos \left({\frac {3 \pi} 2 - \theta}\right)}\)

Tangent is Sine divided by Cosine

\(\ds \)

\(=\)

\(\ds \frac {-\cos \theta} {-\sin \theta}\)

Sine of Three Right Angles and Cosine of Three Right Angles

\(\ds \)

\(=\)

\(\ds \cot \theta\)

Cotangent is Cosine divided by Sine

$\blacksquare$

Also see

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

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