12sin4x

12sin4x is a mathematical expression involving a trigonometric function. It represents the sine of an angle, 4x, multiplied by the constant 12. The sine function, sin(θ), is a fundamental periodic function in trigonometry that relates an angle of a right-angled triangle to the ratio of the length of the side opposite the angle to the length of the hypotenuse. The sine function has a range of values between -1 and 1, inclusive.

In the expression 12sin4x, the argument of the sine function is 4x. This means that the angle is scaled by a factor of 4. This scaling affects the period of the trigonometric function. The period of sin(x) is 2π. For sin(kx), the period is 2π/|k|. Therefore, the period of sin(4x) is 2π/4, which simplifies to π/2. The multiplication by 12 outside the sine function represents an amplitude scaling. This means that the maximum value of 12sin4x will be 12 * 1 = 12, and the minimum value will be 12 * (-1) = -12. The overall range of the expression is therefore [-12, 12]. This type of expression can appear in various contexts, including wave mechanics, signal processing, and physics problems involving oscillations.