Sine of Angle plus Right Angle
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Theorem
- $\sin \paren {x + \dfrac \pi 2} = \cos x$
Proof
\(\ds \sin \paren {x + \frac \pi 2}\) | \(=\) | \(\ds \sin x \cos \frac \pi 2 + \cos x \sin \frac \pi 2\) | Sine of Sum | |||||||||||
\(\ds \) | \(=\) | \(\ds \sin x \cdot 0 + \cos x \cdot 1\) | Cosine of Right Angle and Sine of Right Angle | |||||||||||
\(\ds \) | \(=\) | \(\ds \cos x\) |
$\blacksquare$
Also see
- Cosine of Angle plus Right Angle
- Tangent of Angle plus Right Angle
- Cotangent of Angle plus Right Angle
- Secant of Angle plus Right Angle
- Cosecant of Angle plus Right Angle
Sources
- 1960: Walter Ledermann: Complex Numbers ... (previous) ... (next): $\S 4.5$. The Functions $e^z$, $\cos z$, $\sin z$: $(4.19)$
- 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
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Appendix $12$: Trigonometric formulae: Shifts and periodicity
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Shifts and periodicity