Sine of Angle plus Full Angle

Theorem

$\map \sin {x + 2 \pi} = \sin x$

Corollary

Let $n \in \Z$ be an integer.

Then:

$\map \sin {x + 2 n \pi} = \sin x$

Proof

\(\ds \map \sin {x + 2 \pi}\)

\(=\)

\(\ds \sin x \cos 2 \pi + \cos x \sin 2 \pi\)

Sine of Sum

\(\ds \)

\(=\)

\(\ds \sin x \cdot 1 + \cos x \cdot 0\)

Cosine of Full Angle and Sine of Full Angle

\(\ds \)

\(=\)

\(\ds \sin x\)

$\blacksquare$

Also see

• Cosine of Angle plus Full Angle
• Tangent of Angle plus Full Angle
• Cotangent of Angle plus Full Angle
• Secant of Angle plus Full Angle
• Cosecant of Angle plus Full 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