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 the sine and cosine functions is the set of all real numbers. In interval notation, this is represented as:
- Domain of sine: (−∞,∞)(-\infty, \infty)(−∞,∞)
- Domain of cosine: (−∞,∞)(-\infty, \infty)(−∞,∞)
Explanation:
Both the sine and cosine functions are periodic, meaning they repeat their values in regular intervals. These two trigonometric functions are defined for every real number, making their domains the entire set of real numbers.
- Sine Function: The sine of an angle in a right triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. In the unit circle, the sine function can be interpreted as the y-coordinate of a point on the unit circle corresponding to an angle. Since the unit circle is defined for all angles, there are no restrictions on the values of xxx that can be input into the sine function. As a result, the domain of the sine function is all real numbers, which we write as (−∞,∞)(-\infty, \infty)(−∞,∞).
- Cosine Function: The cosine of an angle is similarly defined in terms of a right triangle as the ratio of the length of the adjacent side to the length of the hypotenuse. In the unit circle, the cosine function is the x-coordinate of a point corresponding to a given angle. Like sine, since the unit circle exists for all angles, the cosine function can take any real number as its input. Therefore, the domain of the cosine function is also (−∞,∞)(-\infty, \infty)(−∞,∞).
In summary, both sine and cosine functions are defined for all real values of the angle, and thus, their domains are the entire set of real numbers. This is a crucial feature of these functions, making them widely applicable in various fields such as physics, engineering, and signal processing.
