‘Where is the tangent function undefined? Choose the correct answer below: Xis an even multiple of 2 Xis an odd multiple of x} {x: xis an even multiple of z} Xis an odd multiple of 2}’
The Correct Answer and Explanation is:
The tangent function is undefined when the cosine of the angle is equal to zero. The tangent of an angle is given by the ratio:tan(x)=sin(x)cos(x)\tan(x) = \frac{\sin(x)}{\cos(x)}tan(x)=cos(x)sin(x)
For tan(x)\tan(x)tan(x) to be undefined, the denominator (cos(x)\cos(x)cos(x)) must be zero. The cosine function equals zero at odd multiples of π2\frac{\pi}{2}2π, i.e., at:x=π2+nπwhere n is any integerx = \frac{\pi}{2} + n\pi \quad \text{where } n \text{ is any integer}x=2π+nπwhere n is any integer
These points correspond to the values where the tangent function has vertical asymptotes and becomes undefined. The cosine function equals zero at values like π2\frac{\pi}{2}2π, 3π2\frac{3\pi}{2}23π, 5π2\frac{5\pi}{2}25π, and so on, which are odd multiples of π2\frac{\pi}{2}2π.
Thus, the correct answer is:
X is an odd multiple of π2\frac{\pi}{2}2π.
Explanation:
The periodicity of the sine and cosine functions is key to understanding where the tangent function becomes undefined. Both the sine and cosine functions repeat every 2π2\pi2π, but the cosine function crosses zero at regular intervals of odd multiples of π2\frac{\pi}{2}2π, which causes the tangent function to be undefined at these points.
For example, at x=π2x = \frac{\pi}{2}x=2π, the cosine of the angle is zero, making tan(x)\tan(x)tan(x) undefined. This same pattern repeats at x=3π2x = \frac{3\pi}{2}x=23π, x=5π2x = \frac{5\pi}{2}x=25π, and so on, where the cosine function again equals zero.
Therefore, the tangent function is undefined at these odd multiples of π2\frac{\pi}{2}2π.
