Question

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

Answer 1

Hint: We will be using the concept of trigonometric functions to solve the problem. We will be using the fact that $\cos 60{}^\circ =\dfrac{1}{2}$and the given formula that \[\cos A=\sqrt{\dfrac{1+\cos 2A}{2}}\] to solve the problem.

Complete step-by-step answer:
Now, we have been given that the value of $\cos 60{}^\circ =\dfrac{1}{2}$ and the given formula that \[\cos A=\sqrt{\dfrac{1+\cos 2A}{2}}\] to find the value of $\cos 30{}^\circ $.

Now, let us take the value of angle $2A=60{}^\circ $. So, we have been given that,

$\cos 2A=\cos 60{}^\circ =\dfrac{1}{2}................\left( 1 \right)$

Also, we have been given that,

\[\cos A=\sqrt{\dfrac{1+\cos 2A}{2}}..............\left( 2 \right)\]

So, we substitute the value of $\cos 2A=\dfrac{1}{2}$ from (1) in (2). So, we have,

$\begin{align}

  & \cos A=\sqrt{\dfrac{1+\dfrac{1}{2}}{2}} \\

 & =\sqrt{\dfrac{3}{4}} \\

 & \cos A=\dfrac{\sqrt{3}}{2} \\

\end{align}$

Now, we know that the value of $A=30{}^\circ $. Therefore, we have the value of $\cos

30{}^\circ =\dfrac{\sqrt{3}}{2}$.


Note: To solve these type of question it is important to note that we have used the value of $\cos 60{}^\circ =\dfrac{1}{2}$ and the formula given to us that $\cos A=\sqrt{\dfrac{1+\cos 2A}{2}}$ to find the value of $\cos 30{}^\circ $.