What are the domain and the range of y = sin x?
The Correct Answer and Explanation is:
The function y=sin(x)y = \sin(x)y=sin(x) is one of the most fundamental trigonometric functions. It is defined for all real numbers, meaning its domain is the entire set of real numbers, R\mathbb{R}R. The sine function does not have any restrictions on the input values of xxx, so we can say that the domain of y=sin(x)y = \sin(x)y=sin(x) is:Domain:(−∞,∞)\text{Domain:} \quad (-\infty, \infty)Domain:(−∞,∞)
Explanation of Domain
The sine function is periodic, and it repeats its values in a regular cycle as xxx increases or decreases. This periodicity means that no matter how large or small xxx becomes, the sine function will always produce a valid output. The sine function is defined for every real number, so there are no limitations on the values of xxx.
Range of y=sin(x)y = \sin(x)y=sin(x)
The sine function only takes values between -1 and 1, inclusive. This means that for every input xxx, the output y=sin(x)y = \sin(x)y=sin(x) will always lie within the interval [−1,1][-1, 1][−1,1]. Mathematically, we express this as:Range:[−1,1]\text{Range:} \quad [-1, 1]Range:[−1,1]
Explanation of Range
The sine function oscillates between -1 and 1 as xxx changes. The maximum value of sin(x)\sin(x)sin(x) is 1, which occurs at x=π2+2kπx = \frac{\pi}{2} + 2k\pix=2π+2kπ for any integer kkk. The minimum value of sin(x)\sin(x)sin(x) is -1, which occurs at x=3π2+2kπx = \frac{3\pi}{2} + 2k\pix=23π+2kπ for any integer kkk. This means that no matter the value of xxx, the sine of that number will never exceed 1 or go below -1.
Thus, the domain of y=sin(x)y = \sin(x)y=sin(x) is all real numbers, and the range is the closed interval from -1 to 1.
