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Question: How do you find \(\operatorname{arccot} ( - 2)? \)...

How do you find arccot⁡(−2)?\operatorname{arccot} ( - 2)?

Explanation

Solution

Hint: First consider the result to be xx and then write the inverse trigonometric function in trigonometric function and convert the cotangent into tangent with the help of trigonometric identity between them that is cotangent and tangent are the multiplicative inverse of each other and then find the tangent inverse of the argument on the calculator and hence you get the result.

Complete step by step solution: In order to find the value of arccot⁡(−2)\operatorname{arccot} ( - 2) we will first convert it into tangent inverse function for which we have to first consider the result to be something
Let us consider arccot⁡(−2)=x\operatorname{arccot} ( - 2) = x
Therefore it can be written as cot⁡x=−2\cot x = - 2
Since both are inverse functions of each other.
Now from the trigonometric identity between tangent and cotangent we know that both are
multiplicative inverse of each other
⇒tan⁡xcot⁡x=1 ∴cot⁡x=1tan⁡x  \Rightarrow \tan x\cot x = 1 \\\ \therefore \cot x = \dfrac{1}{{\tan x}} \\\
So replacing cot⁡x  with  1tan⁡x\cot x\;{\text{with}}\;\dfrac{1}{{\tan x}} we will get
⇒cot⁡x=−2 ⇒1tan⁡x=−2 ⇒tan⁡x=−12  \Rightarrow \cot x = - 2 \\\ \Rightarrow \dfrac{1}{{\tan x}} = - 2 \\\ \Rightarrow \tan x = - \dfrac{1}{2} \\\
Therefore, with the help of inverse function value of argument (x)(x) will be given as
⇒x=tan⁡−1(−12)\Rightarrow x = {\tan ^{ - 1}}\left( { - \dfrac{1}{2}} \right)
Now finding this value on a scientific calculator we will get the value of x=−26.570x = - {26.57^0}
The resultant value is the principal value of the given problem but if we want to write the result within the interval [0,  3600]\left[ {0,\;{{360}^0}} \right] we have to do the following steps
Since we know that value of cotangent is negative in the second and fourth quadrant therefore it can be written as
x=1800−26.570  and  3600−26.570 x=153.430  and  333.430  x = {180^0} - {26.57^0}\;{\text{and}}\;{360^0} - {26.57^0} \\\ x = {153.43^0}\;{\text{and}}\;{333.43^0} \\\
We got two values in the interval [0,  3600]\left[ {0,\;{{360}^0}} \right]

Note: The above results are written in degrees, they can be converted into radians with the help of following conversion formula:
x  degrees=π180radiansx\;{\text{degrees}} = \dfrac{\pi }{{180}}{\text{radians}}
Arc in the prefix of “arccot” represents the inverse function of cotangent.