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Formula for the sine of the double angle
The formula for the sine of the double angle is as follows:
sin(2x) = 2senx ⋅ cosx
In other words, the sine of the double angle is equal to twice the sine: of the angle for its cosine. For the sake of clarity, here are some practical examples.
Exercises of the sine of the double angle
We are going to see a first exercise in which we are going to calculate the sine of the double angle of 30º. Therefore, we apply the formula we have seen in the previous point and we have that:
sin(2⋅ 30º) = 2sen 30º ⋅ cos 30º = 2 ⋅ 0.5 ⋅ 0.5 ⋅ 0.866 = 0.866
For the avoidance of doubt, we will calculate the sine of the double angle of 60º. Following the procedure exactly as before, we will be left with the following:
sin(2 ⋅ 60º) = 2sen 60º ⋅ cos 60º = 2 ⋅ 0.866 ⋅ 0.500 = 0.866
As you can see, calculate the sine of the double angle does not involve any difficulty beyond remembering the formula and operating correctly. If you have any questions, leave us a comment and we will help you.
Derivative of the sine of the double angle
The derivative of the sine of the double angle is as follows:
f'(sin(2x)) = x' cosx = 2cos(2x)
Integral of the sine of the double angle
If what we want is to calculate the integral, the following formula is the one to use:
∫ sin 2x = -2cos (2x)
Demonstration of the sine of the double angle
The demonstration of the double angle is very simple. We start from the following property of the sine in which:
sin (a + b) = sin(a) ⋅ cos (b) + cos(a) ⋅ sin(b)
But it should be noted that in the case of the double angle a = b, therefore:
sin (a + a) = sin(a) ⋅ cos (a) + cos(a) ⋅ sin(a) = 2sen(a)cos(a)
In this simple way this trigonometric ratio is demonstrated.
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2 thoughts on “Seno del ángulo doble”
hello I have an example of a double angle that tells me: Determine the formula for
2a = a + a
Hola tengo un ejercicio en el que no entiendo el enunciado:
Sea cos a= -5/3 si a pertenece al III cuadrante hallar el sen 2a y el cos 4a (aplicar funciones trigonométricas del ángulo doble y mitad)