What is the domain of the cosine function?
Choose the correct answer below.
A. The domain of the cosine function is the set of all real numbers, except odd integer multiples of (180°).
B. The domain of the cosine function is the set of all real numbers. The domain of the cosine function is the set of all real numbers,
C. except odd integer multiples of ??2(90°).
D. The domain of the cosine function is except integer multiples of (180°)
The Correct Answer and Explanation is :
The correct answer is B. The domain of the cosine function is the set of all real numbers.
Explanation:
The cosine function, denoted as ( \cos(x) ), is a trigonometric function that describes the relationship between the angle of a right triangle and the ratio of the adjacent side to the hypotenuse. However, in the context of the cosine function, we typically refer to ( \cos(x) ) where ( x ) represents an angle measured in either radians or degrees.
Unlike other trigonometric functions like tangent, which can have specific undefined points (such as odd integer multiples of ( 90^\circ ) or ( \frac{\pi}{2} )), the cosine function is defined for all real numbers. This means that for any real value of ( x ), ( \cos(x) ) will yield a finite value. The cosine function is periodic, meaning it repeats its values in a regular cycle.
In terms of the domain, there are no restrictions for the cosine function. It can accept any real number as input. Specifically:
- The domain of ( \cos(x) ) is all real numbers, denoted as ( (-\infty, \infty) ).
- There are no values of ( x ) for which the cosine function is undefined.
In contrast, other trigonometric functions, such as tangent or secant, have restrictions. For instance, the tangent function is undefined at odd integer multiples of ( 90^\circ ) (or ( \frac{\pi}{2} ) radians), and the secant function is undefined at even integer multiples of ( 90^\circ ) (or integer multiples of ( \frac{\pi}{2} )).
Key Points:
- The cosine function does not have any restrictions on its domain.
- The domain of ( \cos(x) ) is all real numbers.
- ( \cos(x) ) is periodic with a period of ( 360^\circ ) (or ( 2\pi ) radians), meaning it repeats its values every full rotation.
Thus, B is the correct answer.