Question

Find the value of $\cos \left( {\dfrac{{2\pi }}{3}} \right)$?

Answer 1

Here in this question, we have to find the exact value of given trigonometric function by using the cosine sum or difference identity. First rewrite the given angle in the form of addition or difference, then the standard trigonometric formula cosine sum i.e., $cos\,(A + B)$or cosine difference i.e., $cos\,(A - B)$ identity defined as $cos\,A.cos\,B - sin\,A.sin\,B$ and $cos\,A.cos\,B + sin\,A.sin\,B$using one of these we get required value.

Complete step-by-step solution:
To evaluate the given question by using a formula of cosine addition defined as the cosine addition formula calculates the cosine of an angle that is either the sum or difference of two other angles. It arises from the law of cosines and the distance formula. By using the cosine addition formula, the cosine of both the sum and difference of two angles can be found with the two angles' sines and cosines.
Consider the given function
$\cos \left( {\dfrac{{2\pi }}{3}} \right)$-------(1)
The angle $\dfrac{{2\pi }}{3}$ can be written as $\pi - \dfrac{\pi }{3}$, then
Equation (1) becomes
$\Rightarrow \,\cos \left( {\pi - \dfrac{\pi }{3}} \right)$ ------(2)
Apply the trigonometric cosine identity of difference cos\,(A - B) = $$$$cos\,A.cos\,B + sin\,A.sin\,B.
Here $A = \,\pi$ and $B = \,\dfrac{\pi }{3}$
Substitute A and B in formula then
$\Rightarrow \,\cos \left( {\pi - \dfrac{\pi }{3}} \right) = cos\,\pi .cos\,\dfrac{\pi }{3} + sin\,\pi .sin\,\dfrac{\pi }{3}$
By using specified cosine and sine angle i.e., $cos\,\,\dfrac{\pi }{3} = \dfrac{1}{2}$, $cos\,\,\pi = - 1$, $sin\,\dfrac{\pi }{3} = \dfrac{{\sqrt 3 }}{2}$ and $sin\,\pi = 0$
On, Substituting the values, we have
$\Rightarrow \,\cos \left( {\pi - \dfrac{\pi }{3}} \right) = \dfrac{1}{2}.\left( { - 1} \right) + \dfrac{{\sqrt 3 }}{2}.\left( 0 \right)$
On simplification we get
$\Rightarrow \,\cos \left( {\pi - \dfrac{\pi }{3}} \right) = - \dfrac{1}{2} + 0$
$\Rightarrow \,\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}$

Hence, the exact functional value of $\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}$.

Note: Simply this can also be solve by using a ASTC rule i.e.,
$\Rightarrow \,\,\cos \left( {\dfrac{{2\pi }}{3}} \right) = \cos \left( {\pi - \dfrac{\pi }{3}} \right)$
By using the ASTC rule of trigonometry, the angle $\pi - \dfrac{\pi }{3}$ or angle $180 - \theta$ lies in the second quadrant. cosine function are negative here, hence the angle must in negative, then
$\Rightarrow \,\,\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \cos \left( {\dfrac{\pi }{3}} \right)$
$\Rightarrow \,\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}$
While solving this type question, we must know about the ASTC rule.
And also know the cosine sum or difference identity, for this we have a standard formula. To find the value for the trigonometry function we need the table of trigonometry ratios for standard angles.