Question

How do you find $\cos \left( \dfrac{x}{2} \right)$ if $\cos \left( x \right)=-\dfrac{31}{49}$ using the half-angle identity?

Answer 1

In order to find the solution of the given question that is to find $\cos \left( \dfrac{x}{2} \right)$ if $\cos \left( x \right)=-\dfrac{31}{49}$ using the half-angle identity apply the formula of trigonometric half-angle identity that is $\cos \left( x \right)=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1$ then substitute the given value which is $\cos \left( x \right)=-\dfrac{31}{49}$ in this formula and find the value of the remaining unknown term in the equation that is $\cos \left( \dfrac{x}{2} \right)$ .

Complete step by step solution:
According to the question, given value in the question is as follows:
$\cos \left( x \right)=-\dfrac{31}{49}$
We have to find the value of $\cos \left( \dfrac{x}{2} \right)$ using the above value and half-angle identity.
Now we will apply the formula of trigonometric half-angle identity that is $\cos \left( x \right)=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1$ and substitute the given value then we will have:
$\Rightarrow -\dfrac{31}{49}=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1$
After this add $1$ to both the sides of the above equation, we will have:
$\Rightarrow -\dfrac{31}{49}+1=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)-1+1$
Now simplify the above equation with the help of addition, we will have:
$\Rightarrow 1-\dfrac{31}{49}=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)$
After this take the LCM of the terms in the left-hand side of the above equation, we will have:
$\Rightarrow \dfrac{18}{49}=2{{\cos }^{2}}\left( \dfrac{x}{2} \right)$
Now divide $2$ from both sides of the given equation, we will have:
$\Rightarrow \dfrac{18}{49\times 2}=\dfrac{2{{\cos }^{2}}\left( \dfrac{x}{2} \right)}{2}$
After this simplify the above equation with the help of division, we will have:
$\Rightarrow \dfrac{9}{49}={{\cos }^{2}}\left( \dfrac{x}{2} \right)$
We can rewrite the above equation as follows:
$\Rightarrow {{\cos }^{2}}\left( \dfrac{x}{2} \right)=\dfrac{9}{49}$
Now take square root from both the sides of the equation, we will have:
$\Rightarrow \cos \left( \dfrac{x}{2} \right)=\pm \dfrac{3}{7}$
Therefore, the values of $\cos \left( \dfrac{x}{2} \right)$ are $\dfrac{3}{7}$ and $-\dfrac{3}{7}$.

Note: Students make mistakes while not considering both the negative and positive value when they take the square root of a number. Students tend to consider the positive root only which leads to an incomplete answer.