Question

How do you find the exact values of $\cos \dfrac{\pi }{8}$ using the half-angle formula?

Answer 1

Here, we will first rewrite the given angle in such a way that it is in the form of a half-angle. Then we will use the half-angle formula and a standard angle value of cosine to simplify the equation. We will then use the basic mathematical operation to simplify the equation further to find the required value.

Formula used:
The cosine of a half angle is given by the formula $\cos \dfrac{A}{2} = \pm \sqrt {\dfrac{{\cos A + 1}}{2}}$.

Complete step-by-step solution:
We will use the half-angle formula for cosine to find the value of $\cos \dfrac{\pi }{8}$.
We can rewrite the given angle as a half-angle.
Rewriting the given expression, we get
$\cos \dfrac{\pi }{8} = \cos \left( {\dfrac{1}{2} \times \dfrac{\pi }{4}} \right)$
The cosine of a half angle is given by the formula $\cos \dfrac{A}{2} = \pm \sqrt {\dfrac{{\cos A + 1}}{2}}$.
Substituting $A = \dfrac{\pi }{4}$ in the half angle formula, we get the equation
$\Rightarrow \cos \left( {\dfrac{{\dfrac{\pi }{4}}}{2}} \right) = \pm \sqrt {\dfrac{{\cos \dfrac{\pi }{4} + 1}}{2}}$
Simplifying the L.H.S., we get
$\Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\cos \dfrac{\pi }{4} + 1}}{2}}$
We will simplify the expression on the right-hand side to get the required value of $\cos \dfrac{\pi }{8}$.
The cosine of an angle measuring $\cos \dfrac{\pi }{4}$ is equal to $\dfrac{1}{{\sqrt 2 }}$.
Substituting $\cos \dfrac{\pi }{4} = \dfrac{1}{{\sqrt 2 }}$ in the equation $\cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\cos \dfrac{\pi }{4} + 1}}{2}}$, we get
$\Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\dfrac{1}{{\sqrt 2 }} + 1}}{2}}$
Taking the L.C.M. of the terms in the numerator, we get
$\Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\dfrac{{1 + \sqrt 2 }}{{\sqrt 2 }}}}{2}}$
Simplifying the expression, we get
$\Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{1 + \sqrt 2 }}{{2\sqrt 2 }}}$
The denominator is a radical expression. We will rationalize the denominator.
Multiplying and dividing the fraction inside the radical sign by $\sqrt 2$, we get
$\Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{1 + \sqrt 2 }}{{2\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}}$
Multiplying the terms using the distributive law of multiplication, we get
$\begin{array}{l} \Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\sqrt 2 + 2}}{{2 \times 2}}} \\\ \Rightarrow \cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\sqrt 2 + 2}}{4}} \end{array}$
The number 4 is a perfect square.
Taking the perfect square outside the radical sign, we get
$\therefore \cos \dfrac{\pi }{8}=\dfrac{\sqrt{\sqrt{2}+2}}{2}$

Therefore, we get the required value of $\cos \dfrac{\pi }{8}$ as $\dfrac{{\sqrt {\sqrt 2 + 2} }}{2}$.

Note:
The half angle formula for cosine is derived from the double angle formula for cosine. The cosine of a double angle is given by the formula $\cos 2A = 2{\cos ^2}A - 1$. We can observe that by rearranging the double angle formula, we get the equation $\cos A = \sqrt {\dfrac{{\cos 2A + 1}}{2}}$. This can be written as the half angle formula $\cos \dfrac{A}{2} = \sqrt {\dfrac{{\cos A + 1}}{2}}$.
We have considered $\cos \dfrac{\pi }{8} = \sqrt {\dfrac{{\cos \dfrac{\pi }{4} + 1}}{2}}$ and not $\cos \dfrac{\pi }{8} = - \sqrt {\dfrac{{\cos \dfrac{\pi }{4} + 1}}{2}}$ because the angle $\dfrac{\pi }{8}$ lies in the first quadrant, and the trigonometric ratio of cosine of any angle in the first quadrant is positive.