Explained: "How to find the exact values of cos..."

amamnebrajqx

Answered question

2023-02-11

How to find the exact values of cos 22.5 degrees using the half angle formula?

Marley Rowe

Beginner2023-02-12Added 6 answers

It is possible to derive the cosine half angle identity. (since I don't recall it off-hand):

\cos^{2} (x) = \frac{1 + \cos (2 x)}{2}

By inference:

\cos^{2} (\frac{x}{2}) = \frac{1 + \cos x}{2}

Square root to get:

\cos (\frac{x}{2}) = \pm \sqrt{\frac{1 + \cos x}{2}}

+

if quadrant I or IV

-

if quadrant II or III

{22.5}^{o}

is quadrant I, so it is positive.

\cos (\frac{45^{o}}{2}) = \sqrt{\frac{1 + {\cos 45}^{o}}{2}}

= \sqrt{\frac{1 + (\frac{\sqrt{2}}{2})}{2}}

= \sqrt{\frac{(\frac{2 + \sqrt{2}}{2})}{2}}

= \sqrt{\frac{2 + \sqrt{2}}{4}}

= \frac{\sqrt{2 + \sqrt{2}}}{2}

\approx 0.9238795

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