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Cosine Function

Plots

Plots

Plots

Alternate form

e^(-i x)/2 + e^(i x)/2

Roots

x = π n - π/2, n element Z

Properties as a real function

R (all real numbers)

{y element R : -1<=y<=1}

periodic in x with period 2 π

even

Series expansion at x = 0

1 - x^2/2 + x^4/24 + O(x^6)
(Taylor series)

Derivative

d/dx(cos(x)) = -sin(x)

Indefinite integral

integral cos(x) dx = sin(x) + constant

Identities

cos(x) = (-1)^m cos(m π + x) for m element Z

cos(x) = -1 + 2 cos^2(x/2)

cos(x) = 1 - 2 sin^2(x/2)

cos(x) = cos(x/3) (-1 + 2 cos((2 x)/3))

cos(x) = cos^2(x/2) - sin^2(x/2)

cos(x) = 1/2 (cos(b - x) + cos(b + x)) sec(b)

cos(x) = 1/2 csc(b) (sin(b - x) + sin(b + x))

cos(x) = cos(a) + 2 sin((a - x)/2) sin((a + x)/2)

Global maxima

max{cos(x)} = 1 at x = 2 π n for integer n

Global minima

min{cos(x)} = -1 at x = 2 π n - π for integer n

min{cos(x)} = -1 at x = 2 π n + π for integer n

Alternative representations

cos(x) = cosh(i x)

cos(x) = 1/sec(x)

cos(x) = cosh(-i x)

Series representations

cos(x) = sum_(k=0)^∞ ((-1)^k x^(2 k))/((2 k)!)

cos(x)∝( sum_(k=0)^∞ (-1)^k (d^(1 + 2 k) δ(x))/(dx^(1 + 2 k)))/θ(x)

cos(x) = - sum_(k=0)^∞ ((-1)^k (-π/2 + x)^(1 + 2 k))/((1 + 2 k)!)

Integral representations

cos(x) = 1 - x integral_0^1 sin(t x) dt

cos(x) = - integral_(π/2)^x sin(t) dt

cos(x) = -i/(2 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) e^(s - x^2/(4 s))/sqrt(s) ds for γ>0

cos(x) = -i/(2 sqrt(π)) integral_(-i ∞ + γ)^(i ∞ + γ) (4^s x^(-2 s) Γ(s))/Γ(1/2 - s) ds for (0<γ<1/2 and x>0)

Definite integral

integral_0^(π/2) cos(x) dx = 1

Definite integral mean square

integral_0^(2 π) (cos^2(x))/(2 π) dx = 1/2 = 0.5

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