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