Question

Solve the following equation:
\[\cot x=-1\]

Answer 1

Hint: - In this type of question, we can use the trigonometric relations to convert the cosec, sec, tan and cot functions in terms of sin and cos functions which can be done using the following relations
\[\begin{align}
 (I)& \tan x=\dfrac{\sin x}{\cos x} \\
 (II)& \cot x=\dfrac{\cos x}{\sin x} \\
 (III)& \cos ecx=\dfrac{1}{\sin x} \\
 (IV)& \sec x=\dfrac{1}{\cos x} \\
\end{align}\]
The general equation for a cot function is mentioned as follows
\[\begin{align}
  & \cot \theta =\cot x \\
 \Rightarrow & \theta =n\pi +x \\
 & \left( Where\ n\in I\ and\ x\in \left( 0,\pi \right) \right) \\
\end{align}\]
Complete step-by-step answer:
As mentioned in the question, we have to find the value of \[\theta \] .
Now, using the formula that is given in the hint, we can write the following expression and that is
\[\begin{align}
  & \cot \theta =-1 \\
 \Rightarrow & \cot \theta =\cot \left( \dfrac{3\pi }{4} \right)[\because \cot \left( \dfrac{3\pi }{4} \right) = -1] \\
 \Rightarrow & \theta =n\pi +\dfrac{3\pi }{4} \\
 & \left( Where\ n\in I\ and\ x\in \left( 0,\pi \right) \right) \\
\end{align}\]
Hence, this is the value of \[\theta \] that is required.

Note: - The students can make an error if they don’t know how to express the cot function and get to the general equation for the angle on which cot function has been applied.
Knowing the trigonometric relations that are mentioned in the hint to convert the given expression in terms of different trigonometric functions, is very important to solve the question.
Another method to do this question is by taking the reciprocal of both sides. By doing this, we can get the equation as \[\tan \theta =-1\].
 Now, we can use the general equation for tan function to get the answer.