Sine of Three Right Angles less Angle

Theorem

$\map \sin {\dfrac {3 \pi} 2 - \theta} = -\cos \theta$

where $\sin$ and $\cos$ are sine and cosine respectively.

Proof

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

\(=\)

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

Sine of Difference

\(\ds \)

\(=\)

\(\ds \paren {-1} \times \cos \theta - 0 \times \sin \theta\)

Sine of Three Right Angles and Cosine of Three Right Angles

\(\ds \)

\(=\)

\(\ds -\cos \theta\)

$\blacksquare$

Also see

• 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
• 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