State the domain of the sine and cosine functions. (Enter your answers using interval notation.) domain of sine domain of cosine
The Correct Answer and Explanation is:
The domain of both the sine and cosine functions is all real numbers. This means that for any real number input, both sine and cosine functions are defined.
In interval notation, the domain of both the sine and cosine functions is written as:(−∞,∞)(-\infty, \infty)(−∞,∞)
Explanation
The sine and cosine functions are periodic trigonometric functions. These functions describe the relationship between the angles of a right triangle and the ratio of two of its sides. Specifically, the sine of an angle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the adjacent side to the hypotenuse.
Both of these functions are defined for any real number input. This means that you can substitute any real number for the angle in either the sine or cosine functions, and they will produce a valid result. This is due to their periodic nature, as they repeat their values over regular intervals. In fact, the sine and cosine functions are periodic with a period of 2π2\pi2π, which means that every 2π2\pi2π radians, the values of sine and cosine repeat.
Since sine and cosine functions do not have any restrictions on the values they can take, such as division by zero or square roots of negative numbers, they are defined for all real numbers. Therefore, the domain of both functions is the set of all real numbers.
To further clarify, both functions are continuous and smooth without any breaks or undefined points. For every real number xxx, you can calculate both sin(x)\sin(x)sin(x) and cos(x)\cos(x)cos(x) with no issues.
In summary, the domain of the sine and cosine functions is (−∞,∞)(-\infty, \infty)(−∞,∞), as they are defined for all real values of the input angle.
