Cosecant of Three Right Angles less Angle

Theorem

$\map \csc {\dfrac {3 \pi} 2 - \theta} = -\sec \theta$

where $\csc$ and $\sec$ are cosecant and secant respectively.

Proof

\(\ds \map \csc {\frac {3 \pi} 2 - \theta}\)

\(=\)

\(\ds \frac 1 {\map \sin {\frac {3 \pi} 2 - \theta} }\)

Cosecant is Reciprocal of Sine

\(\ds \)

\(=\)

\(\ds \frac 1 {-\cos \theta}\)

Sine of Three Right Angles less Angle

\(\ds \)

\(=\)

\(\ds -\sec \theta\)

Secant is Reciprocal of Cosine

$\blacksquare$

Also see

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