4sinx

The expression "4sinx" represents a trigonometric function commonly encountered in mathematics, particularly in calculus, physics, and engineering. It is a scaled version of the basic sine function, denoted as sin(x), where the amplitude (the peak value of the wave) is multiplied by 4. The sine function itself is a periodic function that describes a smooth, oscillating wave with a period of 2π radians (approximately 360 degrees).

In the expression "4sinx," the coefficient 4 modifies the vertical stretch of the sine wave, causing its maximum value to increase from 1 to 4 and its minimum value to decrease from -1 to -4. The horizontal scaling, period, and phase shift remain unchanged unless additional transformations are applied, such as "4sin(bx + c)" where *b* and *c* adjust the period and phase shift, respectively.

This function is widely used in modeling periodic phenomena, such as sound waves, alternating current (AC) circuits, and mechanical vibrations. Its graph is a sine wave oscillating between -4 and 4, repeating every 2π units along the x-axis. The derivative of "4sinx" is "4cosx," and its integral is "-4cosx + C," where *C* represents the constant of integration.

In applied contexts, "4sinx" may appear in Fourier series expansions, where complex waveforms are decomposed into sums of sine and cosine terms. It is also fundamental in trigonometric identities, such as those involving double-angle formulas or sum-to-product conversions. Understanding this function is essential for solving differential equations, analyzing harmonic motion, and interpreting oscillatory behavior in various scientific and engineering disciplines.