sinxcos2x

The expression sin(x)cos(2x) is a trigonometric function involving the sine of an angle *x* and the cosine of twice that angle, *2x*. It is commonly encountered in calculus, physics, and engineering when simplifying or analyzing periodic phenomena. The expression can be rewritten or expanded using trigonometric identities to facilitate further analysis or integration.

One of the most useful identities for simplifying sin(x)cos(2x) is the product-to-sum formula, which converts the product of trigonometric functions into a sum or difference of trigonometric functions. Specifically, the identity for sin(A)cos(B) is given by:

sin(A)cos(B) = ½[sin(A+B) + sin(A−B)].

Applying this to sin(x)cos(2x), where A = x and B = 2x, yields:

sin(x)cos(2x) = ½[sin(x + 2x) + sin(x − 2x)]

= ½[sin(3x) + sin(−x)].

Since sine is an odd function, sin(−x) = −sin(x), so the expression further simplifies to:

sin(x)cos(2x) = ½[sin(3x) − sin(x)].

This form is particularly useful in integration, as it allows the expression to be broken into simpler terms that can be integrated individually. For example, the integral of sin(x)cos(2x) can be computed using this identity, resulting in:

∫sin(x)cos(2x)dx = ½∫[sin(3x) − sin(x)]dx

= ½[−(1/3)cos(3x) + cos(x)] + C,

where C is the constant of integration.

Such transformations are fundamental in solving differential equations, analyzing waveforms, and simplifying complex trigonometric expressions in various scientific and engineering applications.