How to find the sine of the angle?
Sine (sin) is one of the direct trigonometric functions. More information about it can be found in our article What is sine.
Sine of an angle in a right triangle
Before you figure out how to find the sine of the angle, you must decide on the legend. Let in a right triangle:
- α is the acute angle whose sine is to be found;
- c - hypotenuse;
- b - the adjacent leg;
- a - opposite leg.
Then to find the sine of the acute angle of a right triangle, it is enough to calculate the ratio of the length of the opposite leg to the length of the hypotenuse: sin (α) = a / c. It is worth remembering that sin 90 ° is always equal to 1.
Sine of an angle in an arbitrary triangle
It is easiest to find the sine of an angle in an arbitrary triangle using the cosine theorem (cos): the square of the length of any side is equal to the sum of the squares of the lengths of the other two sides minus their doubled products and the cosine of the angle between them.
a² = b² + c² - 2 * b * c * cos (α)
From this formula you can find the cosine: cos (α) = (b² + c² - a²) / (2 * b * c)
And since for the same angle sin (α) ² + cos (α) ² = 1 and this is a constant, we can derive a formula for determining the sine:
sin (α) = √ (1 - cos (α) ²) = √ (1- (b² + c²-a²) ² / (2 * b * c) ²).
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