Question

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

Answer 1

Hint: 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.