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Exact value of cos45

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It's a value you should probably memorize:


\cos45^\circ=\frac{\sqrt2}2=\frac1{\sqrt2}

You can derive it using some trigonometric identities, other known values of cosine, and properties of the cosine function. For example, using the double angle identity for cosine:


\cos^2x=\frac{1+\cos2x}2

If
x=45^\circ, then


\cos^245^\circ=\frac{1+\cos90^\circ}2

and you probably know that
\cos90^\circ=0, so


\cos^245^\circ=\frac12

When we take the square root, we should take the positive root because
\cos x>0 whenever
0^\circ<x<90^\circ:


\cos45^\circ=+√(\frac12)\implies\cos45^\circ=\frac1{\sqrt2}

User Matthijs Van Hest
by
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