Question

How do you simplify \[\cos 2\left( {\dfrac{\pi }{3}} \right)\]?

Answer 1

Hint:In the above question, is based on the concept of trigonometry. The sine, cosine, tangent functions can be solved by using the multiple angle formula which is used inside trigonometric functions. By applying the formula of multiple angles on sine function we can further simplify and get the exact
value.

Complete step by step solution:
Trigonometric function means the function of the angle between the two sides. It tells us the relation between the angles and sides of the right-angle triangle. The trigonometric function having multiple angles is the multiple angle formula. Double and triple angles formulas are there under the multiple angle formulas. Generally, it is written in the form \[\cos (nx)\] where n is a positive integer.

Given is the cosine function,\[\cos 2\left( {\dfrac{\pi }{3}} \right)\]

We need to write this in double angle trigonometric form. Taking 2 inside the bracket and multiplying with angle we get,
\[\cos \left( {\dfrac{{2 \times \pi }}{3}} \right) = \cos \dfrac{{2\pi }}{3}\].
The angle \[\dfrac{\pi }{3}\] is \[{60^ \circ }\]which when multiplied by 2 gives \[{120^ \circ }\].Since 120 lies in the second quadrant the value of cosine trigonometric function is negative.
Therefore, we get the following value.
\[\cos \left( {\dfrac{{2\pi }}{3}} \right) = - \dfrac{1}{2}\]
Now, by applying the multiple angle formula,
\[\cos \left( {2\theta } \right) = {\cos ^2} - {\sin ^2}\theta \]
So, let \[\theta = \dfrac{\pi }{3}\]
By further substituting the value of angle and simplifying it we get,
$
\Rightarrow \cos 2\theta = \cos 2\left( {\dfrac{\pi }{3}} \right) \\
\Rightarrow \cos 2\theta = {\cos ^2}\left( {\dfrac{\pi }{3}} \right) - {\sin ^2}\left( {\dfrac{\pi }{3}}
\right) \\
\Rightarrow \cos 2\theta = {\left( {\dfrac{{\sqrt 3 }}{2}} \right)^2} - {\left( {\dfrac{1}{2}} \right)^2} =
\dfrac{1}{2} \\
$

Therefore, on simplifying we get the above value.

Note: An important thing to note is that the cosine function with angle 60 is positive because it lies in the first quadrant where cosine function is positive whereas the angle with 120 lies in the second quadrant where cosine function is negative.