Question

How do you find the exact value of $\cos \dfrac{{5\pi }}{3}?$

Answer 1

Hint:
Whenever they ask for exact value, we can find the exact value by addition or subtraction of angles of the given function. Let us think of the two angles which gives the answer as the given question after addition or subtraction which is suitable. Then by simplifying those two angles we can arrive at the answer.

Complete step by step solution:
Whenever they ask for exact value, we can find the exact value by addition or subtraction of angles of the given function.
 Let us think of the two angles which give the answer as the given question after doing addition or subtraction.
Now, we can rewrite the given question as below
$\cos \dfrac{{5\pi }}{3} = \cos \left( {\dfrac{{6\pi - \pi }}{3}} \right)$
$ \Rightarrow \cos \dfrac{{5\pi }}{3} = \cos \left( {\dfrac{{6\pi }}{3} - \dfrac{\pi }{3}} \right)$
$ \Rightarrow \cos \dfrac{{5\pi }}{3} = \cos \left( {2\pi - \dfrac{\pi }{3}} \right)$
In the above equation $2\pi $ is nothing but ${360^ \circ }$ as $\pi $ is ${180^ \circ }$ . We know that $\cos (360 - \theta ) = \cos \theta $
So by using this relation we can simplify the above equation as
$ \Rightarrow \cos \dfrac{{5\pi }}{3} = \cos \left( {\dfrac{\pi }{3}} \right)$
Therefore, $ \Rightarrow \cos \dfrac{{5\pi }}{3} = \cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}$
The value of $\cos \left( {\dfrac{\pi }{3}} \right) = \dfrac{1}{2}$ we got from the values of trigonometric functions.
The below table gives the details of the values of trigonometric functions:
\[\begin{array}{*{20}{c}}
  {}&0&{\dfrac{\pi }{6}}&{\dfrac{\pi }{4}}&{\dfrac{\pi }{3}}&{\dfrac{\pi }{2}} \\
  {\sin \theta }&0&{\dfrac{1}{2}}&{\dfrac{1}{{\sqrt 2 }}}&{\dfrac{{\sqrt 3 }}{2}}&1 \\
  {\cos \theta }&1&{\dfrac{{\sqrt 3 }}{2}}&{\dfrac{1}{{\sqrt 2 }}}&{\dfrac{1}{2}}&0 \\
  {\tan \theta }&0&{\dfrac{1}{{\sqrt 3 }}}&1&{\sqrt 3 }&\infty \\
  {\csc \theta }&\infty &2&{\sqrt 2 }&{\dfrac{2}{{\sqrt 3 }}}&0 \\
  {\sec \theta }&1&{\dfrac{2}{{\sqrt 3 }}}&{\sqrt 2 }&2&\infty \\
  {\cot \theta }&\infty &{\sqrt 3 }&1&{\dfrac{1}{{\sqrt 3 }}}&0
\end{array}\]
By referring to these trigonometric values of standard angle we can solve the problems easily.

Note:
Whenever they ask us to find the value or exact value of a function try to reduce it to a simple form by making use of two suitable angles which gives the same angle as that of they have asked, so that by using the predefined values of trigonometric functions we can arrive at the correct answer easily. One thing we need to remember here is values if you don’t know the values of standard functions then it is difficult to solve the problems.