# How do you find the exact value for #cos(arctan(2)) #?

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To find the exact value for ( \cos(\arctan(2)) ), you can use the relationship between trigonometric functions and inverse trigonometric functions.

Since ( \arctan(2) ) represents an angle whose tangent is ( 2 ), we can label this angle as ( \theta ). Therefore, ( \tan(\theta) = 2 ).

Using the definition of tangent as the ratio of the opposite side to the adjacent side in a right triangle, we can construct a right triangle with opposite side ( 2 ) and adjacent side ( 1 ) (because ( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = 2 )).

By the Pythagorean theorem, we can find the length of the hypotenuse of this right triangle:

[ \text{hypotenuse} = \sqrt{\text{opposite}^2 + \text{adjacent}^2} = \sqrt{2^2 + 1^2} = \sqrt{5} ]

Now, knowing the values of the opposite side, adjacent side, and hypotenuse, we can determine the cosine of angle ( \theta ):

[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{1}{\sqrt{5}} ]

Therefore, ( \cos(\arctan(2)) = \frac{1}{\sqrt{5}} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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