Select the answer which is equivalent to the given expression using your calculator. $\newline$ $\cos \left(\arctan \frac{1}{\sqrt{399}}\right)$ $\newline$ $\frac{\sqrt{399}}{20}$ $\newline$ $\frac{20}{\sqrt{399}}$ $\newline$ $\frac{1}{\sqrt{399}}$ $\newline$ $\frac{1}{20}$

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Algebra 2

Multiplication with rational exponents

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Q. Select the answer which is equivalent to the given expression using your calculator.

\newline

\cos \left(\arctan \frac{1}{\sqrt{399}}\right)

\newline

\frac{\sqrt{399}}{20}

\newline

\frac{20}{\sqrt{399}}

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\frac{1}{\sqrt{399}}

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\frac{1}{20}

Understand trigonometric functions: Understand the relationship between the trigonometric functions. $\newline$ The expression $\cos(\arctan(x))$ can be understood by considering a right triangle where $x$ is the ratio of the opposite side to the adjacent side ( $\tan = \frac{\text{opposite}}{\text{adjacent}}$ ). We need to find the cosine of the angle, which is the ratio of the adjacent side to the hypotenuse ( $\cos = \frac{\text{adjacent}}{\text{hypotenuse}}$ ).
Apply relationship to expression: Apply the relationship to the given expression. $\newline$ Let's denote the angle whose $\arctan$ is $\frac{1}{\sqrt{399}}$ as $\theta$ . So, $\tan(\theta) = \frac{1}{\sqrt{399}}$ . In a right triangle, this means the opposite side is $1$ and the adjacent side is $\sqrt{399}$ . We need to find the hypotenuse using the Pythagorean theorem.
Use Pythagorean theorem: Use the Pythagorean theorem to find the hypotenuse. $\newline$ The Pythagorean theorem states that in a right triangle, the square of the hypotenuse $c$ is equal to the sum of the squares of the other two sides $a$ and $b$ . So, $c^2 = a^2 + b^2$ .
Calculate hypotenuse: Calculate the hypotenuse. $\newline$ Let's calculate the hypotenuse $c$ : $\newline$ $c^2 = 1^2 + (\sqrt{399})^2$ $\newline$ $c^2 = 1 + 399$ $\newline$ $c^2 = 400$ $\newline$ $c = \sqrt{400}$ $\newline$ $c = 20$
Find cosine of angle: Find the cosine of the angle. $\newline$ Now that we have the adjacent side $\sqrt{399}$ and the hypotenuse $20$ , we can find $\cos(\theta)$ : $\newline$ $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$ $\newline$ $\cos(\theta) = \frac{\sqrt{399}}{20}$