What is the range of y = sin(x)?

The range of \( y = \sin(x) \) is \([-1, 1]\).

The sine function, denoted as \(\sin(x)\), is a periodic function that oscillates between -1 and 1. This means that for any value of \( x \), the output of \(\sin(x)\) will always lie within this interval. The sine function achieves its maximum value of 1 and its minimum value of -1 at specific points within its cycle. For example, \(\sin(0) = 0\), \(\sin(\pi/2) = 1\), \(\sin(\pi) = 0\), \(\sin(3\pi/2) = -1\), and \(\sin(2\pi) = 0\).

The periodic nature of the sine function means it repeats its values in a regular pattern every \(2\pi\) radians. This is known as the period of the sine function. Within each period, the function smoothly transitions from 0 to 1, back to 0, down to -1, and returns to 0. This smooth, wave-like pattern is called a sinusoidal wave.

Understanding the range of the sine function is crucial when solving trigonometric equations and analysing graphs. It helps you predict the possible values of \(\sin(x)\) and ensures you don't expect results outside the interval \([-1, 1]\). This knowledge is particularly useful in various applications, such as physics, engineering, and even music, where wave patterns and oscillations are common.