Sine function

sin(x), sine function.

Sine definition

In a right triangle ABC the sine of α, sin(α) is defined as the ratio between the side opposite to angle α and the side opposite to the right angle (hypotenuse):

sin α = a / c


a = 3"

c = 5"

sin α = a / c = 3 / 5 = 0.6

Graph of sine


Sine rules

Rule name Rule
Symmetry sin(-θ) = -sin θ
Symmetry sin(90°- θ) = cos θ
Pythagorean identity sin2 α + cos2 α = 1
  sin θ = cos θ × tan θ
  sin θ = 1 / csc θ
Double angle sin 2θ = 2 sin θ cos θ
Angles sum sin(α+β) = sin α cos β + cos α sin β
Angles difference sin(α-β) = sin α  cos β - cos α sin β
Sum to product sin α + sin β = 2 sin [(α+β)/2] cos [(α-β)/2]
Difference to product sin α - sin β = 2 sin [(α-β)/2] cos [(α+β)/2]
Law of sines a / sin α = b / sin β = c / sin γ
Derivative sin' x = cos x
Integral ∫ sin x dx = - cos x + C
Euler's formula sin x = (eix - e-ix) / 2i

Inverse sine function

The arcsine of x is defined as the inverse sine function of x when -1≤x≤1.

When the sine of y is equal to x:

sin y = x

Then the arcsine of x is equal to the inverse sine function of x, which is equal to y:

arcsin x = sin-1(x) = y

See: Arcsin function

Sine table





sin x
-90° -π/2 -1
-60° -π/3 -√3/2
-45° -π/4 -√2/2
-30° -π/6 -1/2
0 0
30° π/6 1/2
45° π/4 2/2
60° π/3 3/2
90° π/2 1

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