Sine of Three Right Angles less Angle
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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