Question

Using the formula, $\cos A = \sqrt {\dfrac{{1 + \cos 2A}}{2}}$ find the value of $\cos {30^ \circ }$, it is being given that $\cos {60^ \circ } = \dfrac{1}{2}$

Answer 1

Hint: To solve this problem we need to have knowledge about trigonometric values. Here let us put $A = {30^ \circ }$ in a given formula and simplify it.

Complete step by step answer:
Here we have to find the value of $\cos {30^ \circ }$
Let us use the given formula $\cos A = \sqrt {\dfrac{{1 + \cos 2A}}{2}}$
Now let us substitute $A = {30^ \circ }$ in above formula where we get
$\Rightarrow \cos {30^ \circ } = \sqrt {\dfrac{{1 + \cos 2 \times {{30}^ \circ }}}{2}}$
$\Rightarrow \cos {30^ \circ } = \sqrt {\dfrac{{1 + \cos {{60}^ \circ }}}{2}}$
Here it is given that $\cos {60^ \circ } = \dfrac{1}{2}$ .Now on substituting the value in the above term we get
$\Rightarrow \cos {30^ \circ } = \sqrt {\dfrac{{1 + \dfrac{1}{2}}}{2}}$
$\Rightarrow \cos {30^ \circ } = \sqrt {\dfrac{3}{4}}$
$\Rightarrow \cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$

Therefore the value of $\cos {30^ \circ } = \dfrac{{\sqrt 3 }}{2}$

NOTE: We have other formulas of $\cos 2A$ as:
$\Rightarrow \cos 2A = {\cos ^2}A - {\sin ^2}A \\\ \Rightarrow \cos 2A = 1 - 2{\sin ^2}A \\\$
If required, we can use these formulas as well. Although in this problem, we are restricted to use a specific formula. So, we don’t have a choice in this problem.