Question

How do you use the half angle formula to find \[\cos 22.5\]?

Answer 1

Hint:The above question is based on the concept of half angle formula from trigonometric function. The sine, cosine, tangent functions can be solved by using the half angle formula which is used inside trigonometric functions. We have to apply the half angle formula of cosine function and reduce it in such a way that it gives a value.

Complete step by step solution:
Trigonometric function means the function of the angle between the two sides. It tells us the relation between the angles and sides of the right-angle triangle.

The trigonometric functions have half angle formulas and using these formulas we can transform an expression with exponents to one without exponents, but whose angles are multiples of the original angle.

So, our first step will be to convert the angle into the multiple of the original angle.

Let \[\dfrac{\theta }{2} = 22.5\]then multiply by 2 on both sides we get,
\[\theta = {45^ \circ }\]we get this angle.

So, using half angle formula,
\[\cos \dfrac{\theta }{2} = \pm \sqrt {\dfrac{{1 + \cos \theta }}{2}} \]

Further by substituting the value,
\[\cos \left( {22.5} \right) = \pm \sqrt {\dfrac{{1 + \cos {{45}^ \circ }}}{2}} \]

But value of \[\cos {45^ \circ } = \dfrac{1}{{\sqrt 2 }}\] so further rationalizing it with \[\sqrt 2 \].

Therefore,\[\cos {45^ \circ } = \dfrac{{\sqrt 2 }}{2}\]

Then substituting the value, we get,
\[
\Rightarrow \cos ({22.5^ \circ }) = + \sqrt {\dfrac{{1 + \dfrac{{\sqrt 2 }}{2}}}{2}} \\
\Rightarrow \cos ({22.5^ \circ }) = \sqrt {\dfrac{{2 + \sqrt 2 }}{4}} = \dfrac{{\sqrt {2 + \sqrt 2 } }}{2}
\\
\]

Therefore, we get the above solution for the trigonometric function of cosine.

Note: An important thing to note is that we consider positive sign because the sign of the function depends on which quadrant the cosine function will lie. Since the half angle which is 22.5 lies in the first quadrant the cosine function will always give a positive value hence, we get the above value.