Question

How do you find the exact value of \[\cos \left( \dfrac{10\pi }{3} \right)\]?

Answer 1

Hint: Divide \[10\pi \] with 3 and write the given angle as the sum of two angles in which one must be in the range \[\left[ 0,\dfrac{\pi }{2} \right]\]. Here, write \[\dfrac{10\pi }{3}=3\pi +\dfrac{\pi }{3}\]. Now, use the property of cosine of an angle given as: - \[\cos \left( n\pi +\theta \right)=-\cos \theta \], where ‘n’ is any odd integer, to get the simplified form and the answer. Use the value: - \[\cos \left( \dfrac{\pi }{3} \right)=\dfrac{1}{2}\].

Complete step by step answer:
Here, we have been asked to find the exact value of the trigonometric expression: - \[\cos \left( \dfrac{10\pi }{3} \right)\].
Now, we know that \[\dfrac{10\pi }{3}\] is a large angle if converted in degrees. So, we need to find some simpler method to solve the question. On observing the angle \[\dfrac{10\pi }{3}\] we can say that the angle \[10\pi \] is divided by 3, so by this division we can write the given angle as: -
\[\Rightarrow \dfrac{10\pi }{3}=\dfrac{3\times \left( 3\pi \right)+\pi }{3}\]
\[\Rightarrow \dfrac{10\pi }{3}=3\pi +\dfrac{\pi }{3}\] - (1)
We know that cosine function repeats its value after an interval of angle \[2\pi \]. Let us draw the graph of cosine function.

From the above graph we can clearly see that the values of the cosine function starts repeating itself after an interval of \[2\pi \]. Also, we can see that the sign of this function is reversed after an interval of \[\pi \]. So, let us use this property to get our answer. From equation (1), we have,
\[\Rightarrow \dfrac{10\pi }{3}=3\pi +\dfrac{\pi }{3}\]
Taking cosine function both the sides, we get,
\[\Rightarrow \cos \dfrac{10\pi }{3}=\cos \left( 3\pi +\dfrac{\pi }{3} \right)\]
The above expression can be written as: -
\[\begin{align}
  & \Rightarrow \cos \dfrac{10\pi }{3}=\cos \left[ 2\pi +\left( \pi +\dfrac{\pi }{3} \right) \right] \\
 & \Rightarrow \cos \dfrac{10\pi }{3}=\cos \left( \pi +\dfrac{\pi }{3} \right) \\
\end{align}\]
Now, \[\left( \pi +\dfrac{\pi }{3} \right)\] lies in the quadrant and we know that the value of cosine of an angle is negative in this quadrant. This can be clearly seen in the graph. Therefore, the value of \[\cos \left( \pi +\dfrac{\pi }{3} \right)\] will be same as \[\cos \dfrac{\pi }{3}\] but the sign will be opposite. So, we have,
\[\Rightarrow \cos \left( \dfrac{10\pi }{3} \right)=-\cos \left( \dfrac{\pi }{3} \right)\]
Substituting \[\cos \left( \dfrac{\pi }{3} \right)=\dfrac{1}{2}\], we get,
\[\Rightarrow \cos \left( \dfrac{10\pi }{3} \right)=-\dfrac{1}{2}\]

Hence, the exact value of \[\cos \left( \dfrac{10\pi }{3} \right)\] is \[-\dfrac{1}{2}\].

Note: You can use the formula: - \[\cos \left( n\pi +\theta \right)=-\cos \theta \] directly to get the answer. Here, ‘n’ must be an odd integer. It does not matter whether ‘n’ is positive or negative because we can use the property \[\cos \left( -x \right)=\cos x\] to get the value of cosine of any angle x. If ‘n’ will be positive then the angle \[\left( n\pi +\theta \right)\] will lie in \[{{3}^{rd}}\] quadrant and if ‘n’ will be negative then the angle \[\left( n\pi +\theta \right)\] can be made to lie in \[{{2}^{nd}}\] quadrant and in both the conditions cosine function is negative. You may note that whatever situations we are taking, the value of \[\theta \] must be considered in the interval \[\left[ 0,\dfrac{\pi }{2} \right]\], i.e., first quadrant.