Sine of Supplementary Angle
Theorem
- $\map \sin {\pi - \theta} = \sin \theta$
where $\sin$ denotes sine.
That is, the sine of an angle equals its supplement.
Proof
| \(\ds \map \sin {\pi - \theta}\) | \(=\) | \(\ds \sin \pi \cos \theta - \cos \pi \sin \theta\) | Sine of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0 \times \cos \theta - \paren {-1} \times \sin \theta\) | Sine of Straight Angle and Cosine of Straight Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds \sin \theta\) |
$\blacksquare$
Also see
- Cosine of Supplementary Angle
- Tangent of Supplementary Angle
- Cotangent of Supplementary Angle
- Secant of Supplementary Angle
- Cosecant of Supplementary Angle
Sources
- 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text V$. Trigonometry: Angles larger than $90 \degrees$: Examples
- 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: Symmetry
- 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): Appendix $14$: Trigonometric formulae: Symmetry