What is the double angle formula for sin(2x)?

The double angle formula for sine states that the sine of double an angle, specifically sin(2x), can be expressed in terms of the sine and cosine of the angle x. The formula is given by:

sin(2x) = 2sin(x)cos(x)

This formula emerges from the angle addition formula for sine, where sin(a + b) = sin(a)cos(b) + cos(a)sin(b). When both a and b are equal to x, it simplifies to:

sin(x + x) = sin(x)cos(x) + cos(x)sin(x),

which simplifies to

sin(2x) = 2sin(x)cos(x).

This relationship is fundamental in trigonometry as it allows for the simplification of expressions involving the sine of double angles and is widely used in various applications including solving equations and analyzing periodic functions.

The other options presented do not accurately represent the sine of double an angle:

• The expression sin^2(x) + cos^2(x) equals 1 for any angle x and does not involve a double angle.

• The expression 2tan(x) relates to the tangent function and does not correspond to the sine function at all.

• The fraction sin(x