‘Which of the following statement is true? A. The domain of cosine function is the set of real numbers. B. The domain of the tangent function is the set of all real numbers_ C The range of sine function is the set of all real numbers_ D. The range of cosine function is the set of all real numbers
The Correct Answer and Explanation is:
The correct statement is A. The domain of the cosine function is the set of real numbers.
Explanation:
Let’s break down each statement to understand why A is the correct answer:
A. The domain of the cosine function is the set of real numbers.
This statement is true. The cosine function, represented as y=cos(x)y = \cos(x)y=cos(x), is defined for all real values of xxx. There is no restriction on the input for cosine, and it can be evaluated for any real number. For example, cos(0)=1\cos(0) = 1cos(0)=1, cos(π/2)=0\cos(\pi/2) = 0cos(π/2)=0, and cos(π)=−1\cos(\pi) = -1cos(π)=−1, among others. Hence, the domain of the cosine function is indeed the set of all real numbers.
B. The domain of the tangent function is the set of all real numbers.
This statement is false. The tangent function, represented as y=tan(x)y = \tan(x)y=tan(x), has vertical asymptotes where the cosine function is zero, since tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x). This happens at values of x=π2+nπx = \frac{\pi}{2} + n\pix=2π+nπ, where nnn is any integer. At these points, tan(x)\tan(x)tan(x) is undefined. Therefore, the domain of the tangent function is not the entire set of real numbers but is restricted to all real numbers except x=π2+nπx = \frac{\pi}{2} + n\pix=2π+nπ.
C. The range of the sine function is the set of all real numbers.
This statement is false. The sine function, y=sin(x)y = \sin(x)y=sin(x), has a restricted range. The values of sine oscillate between -1 and 1 for all real inputs, so its range is [−1,1][-1, 1][−1,1], not the entire set of real numbers.
D. The range of the cosine function is the set of all real numbers.
This statement is false. Similar to the sine function, the cosine function, y=cos(x)y = \cos(x)y=cos(x), also oscillates between -1 and 1. Therefore, its range is [−1,1][-1, 1][−1,1], not all real numbers.
Conclusion:
Thus, A is the only true statement, as the cosine function is defined for all real numbers, and both the sine and cosine functions have a range between -1 and 1, while the tangent function has undefined values at certain points.
