sin(pi*n)

Input

sin(π n)

Plot

Plot
Plot

Values

n | 1 | 3/2 | 2 | 5/2 | 3 sin(π n) | 0 | -1 | 0 | 1 | 0

Alternate form

1/2 i e^(-i π n) - 1/2 i e^(i π n)

Roots

n = m, m element Z

Integer root

n = m, m element Z

Properties as a function

Periodicity

periodic in n with period 2

Parity

odd

Series expansion at n=0

π n - (π^3 n^3)/6 + (π^5 n^5)/120 + O(n^6) (Taylor series)
Big‐O notation »

Derivative

d/dn(sin(π n)) = π cos(π n)

Indefinite integral

integral sin(π n) dn = -cos(π n)/π + constant

Global minima

min{sin(π n)} = -1 at n = 2 n - 1/2 for integer n
min{sin(π n)} = -1 at n = 2 n + 3/2 for integer n

Global maximum

max{sin(π n)} = 1 at n = 2 n + 1/2 for integer n

Alternative representation

sin(π n) = 1/csc(n π)
sin(π n) = cos(π/2 - n π)
sin(π n) = -cos(π/2 + n π)
More information »

Definite integral over a half-period

integral_0^1 sin(n π) dn = 2/π≈0.63662

Definite integral mean square

integral_0^2 1/2 sin^2(n π) dn = 1/2 = 0.5
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