Question

If \[{{180}^{\circ }}<\theta <{{270}^{\circ }}\] , \[\sin \theta =\dfrac{-3}{5},\cot \theta =\dfrac{4}{3}\], then \[\cos \left( \dfrac{\theta }{2} \right)=\] .
A) \[\dfrac{-1}{\sqrt{10}}\]
B) \[\dfrac{1}{\sqrt{10}}\]
C) \[\dfrac{-1}{10}\]
D) \[10\]

Answer 1

HINT: - Before solving this question, we must know about the most important formula that is used in this question which is as follows
\[\cos \theta =2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right)-1\]
(This is the relation of trigonometry that would be used in this question to get to the correct answer)
Also, the other important relation of trigonometric functions is as follows
\[\cot \theta =\dfrac{\cos \theta }{\sin \theta }\]
Firstly, we will find the value of \[\cos \theta \] and then we will use that value to find \[\cos \dfrac{\theta }{2}\] using the above mentioned equations.

Complete step-by-step answer:
As mentioned in the question, we have to find the value of \[\cos \dfrac{\theta }{2}\] .
Now, using the relation given in the hint, we can find the value of \[\cos \theta \] as follows
\[\begin{align}
  & \Rightarrow \cot \theta =\dfrac{\cos \theta }{\sin \theta } \\
 & \Rightarrow \cos \theta =\cot \theta \cdot \sin \theta \\
 & \Rightarrow \cos \theta =\dfrac{4}{3}\cdot \dfrac{-3}{5}=\dfrac{-4}{5} \\
\end{align}\]
(Using the values given in the question)
Now, we can use this obtained value to get the value of \[\cos \dfrac{\theta }{2}\] as follows
\[\begin{align}
  & \Rightarrow \dfrac{-4}{5}=2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right)-1 \\
 & \Rightarrow 2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right)=1-\dfrac{4}{5} \\
 & \Rightarrow 2{{\cos }^{2}}\left( \dfrac{\theta }{2} \right)=\dfrac{1}{5} \\
 & \Rightarrow {{\cos }^{2}}\left( \dfrac{\theta }{2} \right)=\dfrac{1}{10} \\
 & \cos \left( \dfrac{\theta }{2} \right)=\pm \dfrac{1}{\sqrt{10}} \\
\end{align}\]
(By taking square roots on both the sides)
Now, it is given in the question that \[{{180}^{\circ }}<\theta <{{270}^{\circ }}\] , hence, we can write the following
\[{{90}^{\circ }}<\dfrac{\theta }{2}<{{135}^{\circ }}\]
Hence, as we know that in the second quadrant, cos function is negative, hence, as the required angle lies in the second quadrant, therefore the value of \[\cos \dfrac{\theta }{2}\] is \[\dfrac{-1}{\sqrt{10}}\].

NOTE: - The students can make an error if they don’t know the properties that are linked to the trigonometric function that are mentioned in the hint.
We could have also found out the angle directly by using inverse trigonometric functions and hence, we could have got the right answer.
Also, it is important to keep in mind which quadrant is under consideration which is the third quadrant here as it is very important to accurately assess the signs of the trigonometric functions.
Also, in such questions, one must be very careful while doing calculations as there are very high chances that the students might commit a calculation mistake.