The value of cos function when angle of right triangle equals to $45^°$ is called cosine of angle $45$ degrees. It is expressed as $\cos{(45^°)}$ in mathematics as per sexagesimal system.

$\cos{(45^°)} \,=\, \dfrac{1}{\sqrt{2}}$

The value of cos of angle $45$ degrees in fraction is $\dfrac{1}{\sqrt{2}}$ exactly. It is an irrational number and equal to $0.7071067812\ldots$ in decimal form. However, the approximate value of cos of $45$ degrees is taken as $0.7071$ in mathematics. In general, the value of $\cos{(45^°)}$ is called as trigonometric function or trigonometric ratio of standard angle.

$\cos{(45^°)}$ is written as $\cos{\Big(\dfrac{\pi}{4}\Big)}$ in circular system and also written as $\cos{(50^g)}$ in centesimal system alternatively.

$(1) \,\,\,$ $\cos{\Big(\dfrac{\pi}{4}\Big)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,=\,$ $0.7071067812\ldots$

$(2) \,\,\,$ $\cos{(50^g)}$ $\,=\,$ $\dfrac{1}{\sqrt{2}}$ $\,=\,$ $0.7071067812\ldots$

Now, you know the exact value of cos of $45$ degrees in both fraction and decimal form and also know the approximate value. Now, you can learn how to derive the value of $\cos{\Big(\dfrac{\pi}{4}\Big)}$ in trigonometric mathematics.

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