Question

How do you find the exact values of $\cos \left( {67.5} \right)$ using the half angle formula?

Answer 1

By using trigonometric functions, we can apply the trigonometric ratios for the particular angle and find its value, there are trigonometric ratios for different angles. In this question to find the value of $\cos x$ we mainly use the half angle formula of cosine which is given by,
$\cos x = \pm \sqrt {\dfrac{{1 + \cos 2x}}{2}}$,

Complete step-by-step answer:
Half angle formulas allow the expression of trigonometric functions of angles equal to $\dfrac{x}{2}$ in terms of $x$, which can simplify the functions. Half-angle formulas are useful in finding the values of unknown trigonometric functions.
Now given trigonometric ratio is $\cos \left( {67.5} \right)$,
Using half-angle formula of cosine is given by $\cos x = \pm \sqrt {\dfrac{{1 + \cos 2x}}{2}}$ ,
So, here $x = 67.5$,
Now the formula becomes,
\Rightarrow $$$$\cos 67.5 = \pm \sqrt {\dfrac{{1 + \cos 2\left( {67.5} \right)}}{2}},
Now simplifying we get,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{1 + \cos \left( {135} \right)}}{2}}$,
Now splitting the angle we get,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{1 + \cos \left( {180 - 45} \right)}}{2}}$,
Now using the identity$\cos \left( {180 - x} \right) = - \cos x$, we get,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{1 - \cos 45}}{2}}$,
Now we know that$\cos 45 = \dfrac{1}{{\sqrt 2 }}$, so substituting the value in the expression we get,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{1 - \dfrac{1}{{\sqrt 2 }}}}{2}}$,
Now simplifying we get,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{\sqrt 2 - 1}}{{2\sqrt 2 }}}$,
Now rationalising the expression on the right hand side by multiplying and dividing with$\sqrt 2$, we gte,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{\sqrt 2 - 1}}{{2\sqrt 2 }} \times \dfrac{{\sqrt 2 }}{{\sqrt 2 }}}$,
Now simplifying we get,
$\Rightarrow \cos 67.5 = \pm \sqrt {\dfrac{{2 - \sqrt 2 }}{{2 \times 2}}}$,
Now taking the square out of the square root we get,
$\Rightarrow \cos 67.5 = \pm \dfrac{{\sqrt {2 - \sqrt 2 } }}{2}$,
As the given angle 67.5 lies in the first quadrant, the cos value will be positive, so the required value will be$\cos 67.5 = \dfrac{{\sqrt {2 - \sqrt 2 } }}{2}$.

**Final Answer:
$\therefore$The exact value of $\cos \left( {67.5} \right)$ will be equal to $\dfrac{{\sqrt {2 - \sqrt 2 } }}{2}$. **

Note:
By using some half angles formula we can convert an expression with exponents to one without exponents, and whose angles are multiples of the original angle. It is to be noted that we get half angle formulas from double angle formulas. Here are some half angle identities:
$\sin \dfrac{x}{2} = \pm \sqrt {\dfrac{{1 - \cos x}}{2}}$,
$\cos \dfrac{x}{2} = \pm \sqrt {\dfrac{{1 + \cos x}}{2}}$,
$\tan \dfrac{x}{2} = \dfrac{{\sin x}}{{1 + \cos x}} = \dfrac{{1 - \cos x}}{{\sin x}}$.