Question

How do you find the exact value of \[\cos \dfrac{4\pi }{5}\]?

Answer 1

Hint: Write the given angle \[\dfrac{4\pi }{5}\] equal to \[\left( \pi -\dfrac{\pi }{5} \right)\] and use the identity \[\cos \left( \pi -\theta \right)=-\cos \theta \] to simplify. Now, convert the angle \[\dfrac{\pi }{5}\] into degrees using the relation: - \[\pi \] radian = 180 degrees. Finally, use the value \[\cos {{36}^{\circ }}=\dfrac{\sqrt{5}+1}{4}\] to get the answer.

Complete step by step answer:
Here, we have been provided with the trigonometric function \[\cos \dfrac{4\pi }{5}\] and we are asked to calculate its value.
Now, the given angle of the cosine function is \[\dfrac{4\pi }{5}\]. We can write angle as: -
\[\begin{align}
  & \Rightarrow \dfrac{4\pi }{5}=\left( \pi -\dfrac{\pi }{5} \right) \\
 & \Rightarrow \cos \dfrac{4\pi }{5}=\cos \left( \pi -\dfrac{\pi }{5} \right) \\
\end{align}\]
Clearly, we can see that the angle \[\left( \pi -\dfrac{\pi }{5} \right)\] lies in the \[{{2}^{nd}}\] quadrant and in this quadrant cosine function has negative value. So, the value of \[\cos \left( \pi -\dfrac{\pi }{5} \right)\] will be equal to \[\cos \dfrac{\pi }{5}\] but the value will be negative. So, we have,
\[\Rightarrow \cos \left( \dfrac{4\pi }{5} \right)=-\cos \left( \dfrac{\pi }{5} \right)\]
Converting the angle \[\left( \dfrac{\pi }{5} \right)\] from radian to degrees using the relation \[\pi \] radian = 180 degrees, we get,
\[\Rightarrow \] \[\pi \] radian = 180 degrees
Dividing both the sides with 5, we get,
\[\Rightarrow \dfrac{\pi }{5}\] radian = \[\dfrac{180}{5}\] degrees
\[\Rightarrow \dfrac{\pi }{5}\] radian = 36 degrees
So, the expression becomes,
\[\Rightarrow \cos \left( \dfrac{4\pi }{5} \right)=-\cos {{36}^{\circ }}\]
We know that \[\cos {{36}^{\circ }}=\dfrac{\sqrt{5}+1}{4}\], so we have,
\[\Rightarrow \cos \left( \dfrac{4\pi }{5} \right)=-\left( \dfrac{\sqrt{5}+1}{4} \right)\]
Hence, the exact value of \[\cos \left( \dfrac{4\pi }{5} \right)\] is \[-\left( \dfrac{\sqrt{5}+1}{4} \right)\].

Note:
 One may note that we can derive the value of \[\cos {{36}^{\circ }}\]. What we can do is we will assume \[\theta ={{18}^{\circ }}\] and then calculate \[5\theta ={{90}^{\circ }}\]. Now, we will take cosine function on both the sides and write \[5\theta =3\theta +2\theta \] and expand the function. From here we will solve a biquadratic equation in \[\sin \theta \] and get the value of \[\sin {{18}^{\circ }}\]. Finally, using the relation, \[\cos 2\theta =1-2{{\sin }^{2}}\theta \] we will get the value of \[\cos {{36}^{\circ }}\]. But it will be beneficial if we will remember the values of \[\sin {{18}^{\circ }},\cos {{18}^{\circ }},\sin {{36}^{\circ }},\cos {{36}^{\circ }}\] etc. Because it will not be derived everywhere but directly used.