sin 2x
2sinxcosx
In trigonometry, the sine function is denoted by sin(x), where x is an angle measured in radians. The sine of an angle is defined as the ratio of the length of the side opposite to the angle to the length of the hypotenuse in a right-angled triangle.
Now, let’s consider sin(2x), where 2x is an angle that is twice as large as x.
Using the double angle identity, we can express sin(2x) in terms of sin(x) as:
sin(2x) = 2 sin(x) cos(x)
where cos(x) is the cosine of x, which is the ratio of the length of the adjacent side to the length of the hypotenuse in a right-angled triangle.
Alternatively, we can use the sine angle sum formula to express sin(2x) in terms of sin(x) as:
sin(2x) = sin(x + x) = sin(x) cos(x) + cos(x) sin(x) = 2 sin(x) cos(x)
where we have used the fact that sin(x + x) = sin(x)cos(x) + cos(x)sin(x) = 2 sin(x) cos(x).
Therefore, sin(2x) = 2 sin(x) cos(x) for any value of x.
In summary, the sine of twice an angle is equal to twice the product of the sines and cosines of the angle.
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