Question

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

Answer 1

Hint:
The given question is to find out the value of the given trigonometric ratio. Since there are total six trigonometric ratios in trigonometry which are \[\sin \theta ,\cos \theta ,\tan \theta ,\cos ec \theta ,\sec \theta ,\cot \theta \] with an angle \[\theta \]. Also, these six trigonometric ratios have specific or fixed values at some fixed angles which are \[{0^ \circ }, {30^ \circ }, {45^ \circ }, {60^ \circ }\] and \[{90^ \circ }\] values of trigonometric ratios at these angles are fixed. Also, these angles are in degrees. And we also have angles which are in radians and whose values for \[{30^ \circ },{45^ \circ },{60^ \circ }\] and \[{90^ \circ }\] are \[\dfrac{\pi }{6}, \dfrac{\pi }{4}, \dfrac{\pi }{3}{\text{and}}\dfrac{\pi }{2}\] respectively where \[\pi = {180^ \circ }\](in radians).

Complete step by step solution:
The given question is to find out the value of given trigonometric ratios. Since there are six trigonometric ratios whose value for some specific angles is fixed. Since angle is in degree as well as in radians and the fixed values of the trigonometric ratios at certain angles which are \[{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ }\] and \[{90^ \circ }\], where angle is in degree.
Also, angle in radian is symbolized in terms of \[\pi \] where value of \[\pi \] in radians is \[{180^ \circ }\].
Therefore, angles in degrees and radians are \[{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ },{90^ \circ }\] is equivalent to \[0,\dfrac{\pi }{6},\dfrac{\pi }{4},\dfrac{\pi }{3},\dfrac{\pi }{2}\] where \[\pi = {180^ \circ }\].
And, the fixed values for \[\sin \theta {\text{ and }}\cos \theta \] are given as:

angles	\[{0^ \circ }\]	\[{30^ \circ }\]	\[{45^ \circ }\]	\[{60^ \circ }\]	\[{90^ \circ }\]
\[\sin \theta \]	\[\sqrt {\dfrac{0}{4}} = 0\]	\[\sqrt {\dfrac{1}{4}} = \dfrac{1}{2}\]	\[\sqrt {\dfrac{2}{4}} = \sqrt {\dfrac{1}{2}} = \dfrac{1}{{\sqrt 2 }}\]	\[\sqrt {\dfrac{3}{4}} = \dfrac{{\sqrt 3 }}{2}\]	\[\sqrt {\dfrac{4}{4}} = 1\]
\[\cos \theta \]	\[1\]	\[\dfrac{{\sqrt 3 }}{2}\]	\[\dfrac{1}{{\sqrt 2 }}\]	\[\dfrac{1}{2}\]	\[0\]

Value of \[\sin \theta \] are derived by taking \[0,1,2,3,4\].
Dividing all by \[4\] and taking square root and the values obtained are the values of \[\sin \theta \] at angles \[{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ },{90^ \circ }\],
And values of \[\cos \theta \] are derived by taking the inverse of all the values of \[\sin \theta \] for angles \[{0^ \circ },{30^ \circ },{45^ \circ },{60^ \circ }\] and \[{90^ \circ }\].
So, the values become \[\sin {0^ \circ } = \cos {90^ \circ },\sin {30^ \circ } = \cos {60^ \circ },\sin {45^ \circ } = \cos {45^ \circ },\sin {60^ \circ } = \cos {30^ \circ }\] and \[\sin {90^ \circ } = \cos {0^ \circ }\].
Since we want to find out the value of \[\cos \dfrac{\pi }{3}\] which means \[\cos {60^ \circ }\] which is \[\dfrac{1}{2}\].
Therefore, the value of \[\cos \dfrac{\pi }{3}\] is \[\dfrac{1}{2}\].

Note:
The angles in degree as well as in radians are having fixed value for the fixed trigonometric ratio where angles in degree is denoted by a small circle after the angle and in radians is denoted by using \[\pi \] where \[\pi = {180^ \circ }\] which is the relation to convert degree to radians in trigonometry.