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Select the answer which is equivalent to the given expression using your calculator. cos(arctan ((1)/(sqrt399))) (sqrt399)/(20) (20)/(sqrt399) (1)/(sqrt399) (1)/(20)

Select the answer which is equivalent to the given expression using your calculator.\newlinecos⁡(arctan⁡1399) \cos \left(\arctan \frac{1}{\sqrt{399}}\right) \newline39920 \frac{\sqrt{399}}{20} \newline20399 \frac{20}{\sqrt{399}} \newline1399 \frac{1}{\sqrt{399}} \newline120 \frac{1}{20}

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Q. Select the answer which is equivalent to the given expression using your calculator.\newlinecos⁡(arctan⁡1399) \cos \left(\arctan \frac{1}{\sqrt{399}}\right) \newline39920 \frac{\sqrt{399}}{20} \newline20399 \frac{20}{\sqrt{399}} \newline1399 \frac{1}{\sqrt{399}} \newline120 \frac{1}{20}
  1. Understand trigonometric functions: Understand the relationship between the trigonometric functions.\newlineThe expression cos⁡(arctan⁡(x))\cos(\arctan(x)) can be understood by considering a right triangle where xx is the ratio of the opposite side to the adjacent side (tan⁡=oppositeadjacent\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⁡=adjacenthypotenuse\cos = \frac{\text{adjacent}}{\text{hypotenuse}}).
  2. Apply relationship to expression: Apply the relationship to the given expression.\newlineLet's denote the angle whose arctan⁡\arctan is 1399\frac{1}{\sqrt{399}} as θ\theta. So, tan⁡(θ)=1399\tan(\theta) = \frac{1}{\sqrt{399}}. In a right triangle, this means the opposite side is 11 and the adjacent side is 399\sqrt{399}. We need to find the hypotenuse using the Pythagorean theorem.
  3. Use Pythagorean theorem: Use the Pythagorean theorem to find the hypotenuse.\newlineThe Pythagorean theorem states that in a right triangle, the square of the hypotenuse cc is equal to the sum of the squares of the other two sides aa and bb. So, c2=a2+b2c^2 = a^2 + b^2.
  4. Calculate hypotenuse: Calculate the hypotenuse.\newlineLet's calculate the hypotenuse cc:\newlinec2=12+(399)2c^2 = 1^2 + (\sqrt{399})^2\newlinec2=1+399c^2 = 1 + 399\newlinec2=400c^2 = 400\newlinec=400c = \sqrt{400}\newlinec=20c = 20
  5. Find cosine of angle: Find the cosine of the angle.\newlineNow that we have the adjacent side 399\sqrt{399} and the hypotenuse 2020, we can find cos⁡(θ)\cos(\theta):\newlinecos⁡(θ)=adjacenthypotenuse\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}\newlinecos⁡(θ)=39920\cos(\theta) = \frac{\sqrt{399}}{20}

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