Question

Evaluate \[\cos \left( { - \dfrac{{8\pi }}{3}} \right)\] ?

Answer 1

Hint: Use the trigonometric properties to find the value of \[\cos \left( { - \dfrac{{8\pi }}{3}} \right)\].
Since \[\cos ( - x) = \cos x\] so evaluate \[\cos \left( { - \dfrac{{8\pi }}{3}} \right) = \cos \left( {\dfrac{{8\pi }}{3}} \right)\].
Write \[\cos \left( {\dfrac{{8\pi }}{3}} \right) = \cos \left( {2\pi + \dfrac{{2\pi }}{3}} \right)\] .
Then apply the trigonometric formula; \[\cos (2\pi + x) = \cos x\].

Complete step by step answer:
We have to evaluate \[\cos \left( { - \dfrac{{8\pi }}{3}} \right)\]. The value of the negative angle of the cosine function is the same as the positive angle of the cosine function.
\[\cos ( - x) = \cos x\]
\[ \Rightarrow \cos \left( { - \dfrac{{8\pi }}{3}} \right) = \cos \left( {\dfrac{{8\pi }}{3}} \right)\]
Write the angle \[\dfrac{{8\pi }}{3} = 2\pi + \dfrac{{2\pi }}{3}\].
\[ \Rightarrow \cos \left( {\dfrac{{8\pi }}{3}} \right) = \cos \left( {2\pi + \dfrac{{2\pi }}{3}} \right)\]
According to the trigonometric formula; \[\cos (2\pi + x) = \cos x\],
\[ \Rightarrow \cos \left( {2\pi + \dfrac{{2\pi }}{3}} \right) = \cos \left( {\dfrac{{2\pi }}{3}} \right)\]
We can split the angle \[\dfrac{{2\pi }}{3}\] as $\pi - \dfrac{\pi }{3}$ .
\[ \Rightarrow \cos \left( {\dfrac{{2\pi }}{3}} \right) = \cos \left( {\pi - \dfrac{\pi }{3}} \right)\]
Since $\cos (\pi - \theta ) = - \cos \theta $ we get,
\[ \Rightarrow \cos \left( {\pi - \dfrac{\pi }{3}} \right) = - \cos \left( {\dfrac{\pi }{3}} \right)\]
And,
\[ \Rightarrow - \cos \left( {\dfrac{\pi }{3}} \right) = - \dfrac{1}{2}\]

Note: Another Method:
We have to evaluate \[\cos \left( { - \dfrac{{8\pi }}{3}} \right)\].
Write the angle \[ - \dfrac{{8\pi }}{3} = - 2\pi - \dfrac{{2\pi }}{3}\].
\[ \Rightarrow \cos \left( {\dfrac{{8\pi }}{3}} \right) = \cos \left( { - 2\pi - \dfrac{{2\pi }}{3}} \right)\]
According to the trigonometric formula; \[\cos ( - 2\pi - x) = \cos ( - x)\],
\[ \Rightarrow \cos \left( { - 2\pi - \dfrac{{2\pi }}{3}} \right) = \cos \left( { - \dfrac{{2\pi }}{3}} \right)\]
Apply the property, \[\cos ( - x) = \cos x\],
\[ \Rightarrow \cos \left( { - \dfrac{{2\pi }}{3}} \right) = \cos \left( {\dfrac{{2\pi }}{3}} \right)\]
\[ \Rightarrow \cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}\]
The value of \[\cos \left( { - \dfrac{{8\pi }}{3}} \right)\] is \[ - \dfrac{1}{2}\].