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r-ruslan [8.4K]
3 years ago
13

Exact value of cos45

Mathematics
1 answer:
Anit [1.1K]3 years ago
4 0

It's a value you should probably memorize:

\cos45^\circ=\dfrac{\sqrt2}2=\dfrac1{\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=\dfrac{1+\cos2x}2

If x=45^\circ, then

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

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

\cos^245^\circ=\dfrac12

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

\cos45^\circ=+\sqrt{\dfrac12}\implies\cos45^\circ=\dfrac1{\sqrt2}

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