Question

How do you find the two solutions (in radians and in degree) for \[\cot x=-1\] ?

Answer 1

Hint: In the given question we have been asked to find the solution for the given trigonometric expression. First we need to take the inverse of the given expression. Then by using the general solutions, we will solve the expression for the value of ‘x’. To convert the answer in the form of degree, substitute the value of \[\pi ={{180}^{0}}\] .
We can use the trigonometric relations to write the given expressions in different trigonometric functions;
 \[\cot x=\dfrac{1}{\tan x}\]
 \[\tan x=\dfrac{\sin x}{\cos x}\]
 \[\cot x=\dfrac{\cos x}{\sin x}\]
 \[\sec x=\dfrac{1}{\cos x}\]
 \[co\sec x=\dfrac{1}{\sin x}\]

Complete step by step solution:
We have given that,
 \[\cot x=-1\]
Taking the inverse of \[\cot x\] on both the sides,
 \[x={\cot^{-1}} \left( -1 \right)\]
Thus,
Using the trigonometric ratios table;
The exact value of \[{\cot^{-1}} \left( -1 \right)\] is \[\dfrac{3\pi }{4}\] .
Therefore,
In radians;
 \[\Rightarrow x=\dfrac{3\pi }{4}\]
In degrees;
As we know that,
 \[\pi ={{180}^{0}}\]
Substituting the value, we will get
 \[\Rightarrow x=\dfrac{3\times {{180}^{0}}}{4}=\dfrac{{{540}^{0}}}{4}={{135}^{0}}\]
 \[\Rightarrow x={{135}^{0}}\]
Now,
To find the second solution;
As we know that,
The given cotangent function is negative in the second and fourth quadrants.
Subtracting the reference angle \[\pi \] to find the solution in the third quadrant.
Thus,
 \[x=\dfrac{3\pi }{4}-\pi \]
Taking the LCM of the denominators,
 \[x=\dfrac{3\pi -4\pi }{4}=-\dfrac{\pi }{4}\]
Therefore,
In radians,
 \[x=-\dfrac{\pi }{4}\]
In degrees;
As we know that,
 \[\pi ={{180}^{0}}\]
Substituting the value, we will get
 \[\Rightarrow x=-\dfrac{{{180}^{0}}}{4}=-{{45}^{0}}\]
 \[\Rightarrow x=-{{45}^{0}}\]
Hence this is the required answer.

Note: The students can make an error if they don’t know how to express cot function and get to the general equation for the angle on which the 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 questions.