Cosine of Three Right Angles less Angle

Theorem

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

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

Proof

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

\(=\)

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

Cosine of Difference

\(\ds \)

\(=\)

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

Cosine of Three Right Angles and Sine of Three Right Angles

\(\ds \)

\(=\)

\(\ds -\sin \theta\)

$\blacksquare$

Also see

• Sine 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