Cosine of Supplementary Angle
Theorem
- $\map \cos {\pi - \theta} = -\cos \theta$
where $\cos$ denotes cosine.
That is, the cosine of an angle is the negative of its supplement.
Proof
| \(\ds \map \cos {\pi - \theta}\) | \(=\) | \(\ds \cos \pi \cos \theta + \sin \pi \sin \theta\) | Cosine of Difference | |||||||||||
| \(\ds \) | \(=\) | \(\ds \paren {-1} \times \cos \theta + 0 \times \sin \theta\) | Cosine of Straight Angle and Sine of Straight Angle | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\cos \theta\) |
$\blacksquare$
Examples
Cosine of $\theta - 180 \degrees$
- $\map \cos {\theta - 180 \degrees} = -\cos \theta$
Also see
- Sine of Supplementary Angle
- Tangent of Supplementary Angle
- Cotangent of Supplementary Angle
- Secant of Supplementary Angle
- Cosecant of Supplementary 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
- 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