Question

How do you find the exact value of $\cot \dfrac{{7\pi }}{4}?$

Answer 1

First use the relation between tangent and cotangent to convert cotangent into tangent and then convert the argument of the function into principal argument then finally you write the value.The following trigonometric identities will be used:
$\tan x\cot x = 1\;{\text{and}}\;\tan ( - x) = - \tan x$

Complete step by step answer:
In order to find the exact value of the given trigonometric function $\cot \dfrac{{7\pi }}{4}$, we will first convert the cotangent and tangent.From the trigonometric relations, we know that
\tan x\cot x = 1 \\\ \Rightarrow \cot x = \dfrac{1}{{\tan x}} \\\
Using the above conversional formula between tangent and cotangent, converting the given trigonometric function into tangent
$\cot \dfrac{{7\pi }}{4} = \dfrac{1}{{\tan \dfrac{{7\pi }}{4}}}$
Now we will find the value for $\tan \dfrac{{7\pi }}{4}$ and put its value above to get the required answer.To find the value of $\tan \dfrac{{7\pi }}{4}$, we know that the principal argument of tangent function belongs to $\left[ {\dfrac{{ - \pi }}{2},\;\dfrac{\pi }{2}} \right]$ so we will try to shrink the argument into principal argument

We can write $\dfrac{{7\pi }}{4} = 2\pi - \dfrac{\pi }{4} \Rightarrow \tan \dfrac{{7\pi }}{4} = \tan \left( {2\pi - \dfrac{\pi }{4}} \right) = \tan \left( { - \dfrac{\pi }{4} + 2\pi } \right)$
From the periodic property of trigonometric functions, we know that trigonometric values repeat after a fixed interval which is called its period which is equals to $2\pi$ (a complete cycle / period)
So from this property we can write $\tan (x + 2\pi ) = \tan x$
$\Rightarrow \tan \left( { - \dfrac{\pi }{4} + 2\pi } \right) = \tan \left( { - \dfrac{\pi }{4}} \right)$
Now from trigonometric identity of negative arguments, we know that
\tan ( - x) = - \tan x \\\ \Rightarrow \tan \left( { - \dfrac{\pi }{4}} \right) = - \tan \dfrac{\pi }{4} = - 1 \\\
Substituting this value above to get the value of $\cot \dfrac{{7\pi }}{4}$
$\therefore\cot \dfrac{{7\pi }}{4} = \dfrac{1}{{\tan \dfrac{{7\pi }}{4}}} = \dfrac{1}{{ - 1}} = - 1$

Therefore the exact value of $\cot \dfrac{{7\pi }}{4}\;{\text{is}}\; - 1$

Note: We have reduced the given argument into the principal argument because normally students learn the trigonometric values of the principal arguments only. Learning to reduce given argument into principal one is helpful in solving trigonometric equations.Tangent function has principal period $\pi$ radians, but we can use it as $n\pi ,\;{\text{where}}\;n \in I$ depends on the need of it.