Question

How do you find vertical asymptote for $y = \cot (x)$ ?

Answer 1

Hint : A function involving any two variables x and y, can be easily plotted on the graph paper by putting random values of one variable and calculating the value of the other variable. The numbers that can be written in the form of a fraction such that its denominator is not equal to zero and its decimal expansion is terminating and non-repeating are called rational numbers. Rational functions may have an unknown variable or an equation of the unknown variable in the denominator then the values of x at which the denominator comes out to be will give us the vertical asymptote of the function on the graph paper.

Complete step-by-step answer :
In the given question, we can express cotangent as the ratio of cosine and sine, so the values of x at which the sine function is zero will give us the vertical asymptote for $y = \cot (x)$ .
We know that $\cot x = \dfrac{{\cos x}}{{\sin x}}$
$\Rightarrow y = \dfrac{{\sin x}}{{\cos x}}$
The points at which $\sin x$ is zero are –
$\sin x = 0 \\\ \Rightarrow x = 0,\pi ,2\pi .... \\\ \Rightarrow x = n\pi \;$
,where n is an integer.
Hence for $y = \cot (x)$ , vertical asymptote occurs at the $x = n\pi$ where n is an integer.
So, the correct answer is “ $x = n\pi$ ”.

Note : Trigonometric functions are the ratio of two sides of a right-angled triangle and give us the relation between the angles and the sides of a right-angled triangle. Sine, cosine and tangent are the main functions of trigonometry. Cosecant, secant and cotangent are the reciprocals of the sine, cosine and tangent functions respectively, using this definition, we got that $\cot x = \dfrac{1}{{\tan x}} = \dfrac{1}{{\dfrac{{\sin x}}{{\cos x}}}} = \dfrac{{\cos x}}{{\sin x}}$ that led us to the correct answer.