Sine definition
In a right triangle ABC the sine of α, sin(α) is defined as the ratio betwween the side opposite to angle α and the side opposite to the right angle (hypotenuse):
sin α = a / c
Example
a = 5″
c = 10″
sin α = a / c = 5 / 10 = 0.5
Graph of sine
TBD
Sine rules
| Rule name | Rule |
| Symmetry | sin(-θ) = -sin θ |
| Symmetry | sin(90°- θ) = cos θ |
| Pythagorean identity | sin2α + cos2α = 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:
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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
Sine table
| x(o) | x (rad) | sin x |
| -90o | -π/2 | -1 |
| -60o | -π/3 | -√3/2 |
| -45o | -π/4 | -√2/2 |
| -30o | -π/6 | -1/2 |
| 0o | 0 | 0 |
| 30o | π/6 | 1/2 |
| 45o | π/4 | √2/2 |
| 60o | π/3 | √3/2 |
| 90o | π/2 | 1 |
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