Question

How do you evaluate $\cot \left( {\dfrac{{ - 3\pi }}{4}} \right)$

Answer 1

In order to solve this question first we will simplify the angle so that we get the exact value without any complications and after it we will be changing it to tangent as we are more familiar with it and get the final trigonometric term and we will put, the value of that particular angle in tangent form and get the final answer.

Step by step solution:
For solving this question we need to change it in the form of the simplest angle form since the angle is $- \dfrac{{3\pi }}{4}$ in radian so in degrees it will be $- {135^o}$ so it will come in the third quadrant and the value of it will be same as the value of the ${45^o}$.
$\cot \left( {\dfrac{{ - 3\pi }}{4}} \right) = \cot \left( {\dfrac{\pi }{4}} \right)$
Now we will be transforming it in terms of tan because we are more familiar with the tangent trigonometric ratio and generally all the values are in our mind, if it is known cotangent value then you may do and put it directly it will be easy to calculate.
$\cot \left( {\dfrac{\pi }{4}} \right) = \dfrac{1}{{\tan \left( {\dfrac{\pi }{4}} \right)}}$
As we already know the value of $\tan \left( {\dfrac{\pi }{4}} \right) = 1$ so putting this value we will get:
$\dfrac{1}{1} = 1$
So 1 is the final answer.

Note: While solving these types of questions firstly try to analyze where will be after taking the given rotation and in which quadrant we will stay then it will be very easy for us to find the value of the trigonometric ratio.
All the trigonometric ratios are positive in 1st quadrant,
Only sin and cosec are positive in 2nd quadrant,
Only tan and cot are positive in 3rd quadrant,
Only cos and sec are positive in the 4th quadrant.