Secant of Three Right Angles less Angle

Theorem

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

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

Proof

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

\(=\)

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

Secant is Reciprocal of Cosine

\(\ds \)

\(=\)

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

Cosine of Three Right Angles less Angle

\(\ds \)

\(=\)

\(\ds -\csc \theta\)

Cosecant is Reciprocal of Sine

$\blacksquare$

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
• 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