sin 2x
2sinxcosx
The sine of 2x can be expressed as:
sin 2x = 2 sin x cos x
This expression is derived from the double-angle formula for sine, which states that:
sin 2x = 2 sin x cos x
To understand this formula, we need to look at the unit circle. The unit circle is a circle with a radius of 1, centered at the origin of the coordinate plane. The sine of an angle is the y-coordinate of the point on the unit circle that corresponds to that angle, while the cosine of an angle is the x-coordinate of that same point.
Given an angle x, we can find the coordinates of its corresponding point on the unit circle, which are (cos x, sin x). To find the coordinates of the point that corresponds to an angle 2x, we can use the double-angle formula:
sin 2x = 2 sin x cos x
This formula tells us that the sine of twice an angle (2x) is equal to twice the sine of the angle (sin x) times the cosine of the angle (cos x).
This formula is useful in various trigonometric calculations, and it can also help us simplify complex trigonometric expressions.
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