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
Example
a = 3"
c = 5"
sin α = a / c = 3 / 5 = 0.6
Graph of sine
TBD
Sine rules
Rule name | Rule |
---|---|
Symmetry | sin(-θ) = -sin θ |
Symmetry | sin(90°- θ) = cos θ |
Pythagorean identity | sin^{2} α + cos^{2} 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 = (e^{ix} - 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
x (°) |
x (rad) |
sin x |
---|---|---|
-90° | -π/2 | -1 |
-60° | -π/3 | -√3/2 |
-45° | -π/4 | -√2/2 |
-30° | -π/6 | -1/2 |
0° | 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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