Question

current attempt in progress
use the following triangle to find \\( \cos \theta \\). rationalize any denominators containing radicals that you encounter in the answer.
note: enter the exact, fully simplified answer.
\\( \cos \theta = \\)

Explanation:

Step1: Find the hypotenuse

Using the Pythagorean theorem \(c=\sqrt{a^{2}+b^{2}}\), where \(a = 1\) and \(b=3\).
\(c=\sqrt{1^{2}+3^{2}}=\sqrt{1 + 9}=\sqrt{10}\)

Step2: Calculate \(\cos\theta\)

By the definition of cosine in a right - triangle \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\).
The adjacent side to \(\theta\) is \(1\) and the hypotenuse is \(\sqrt{10}\). So \(\cos\theta=\frac{1}{\sqrt{10}}\).

Step3: Rationalize the denominator

Multiply the numerator and denominator by \(\sqrt{10}\): \(\frac{1\times\sqrt{10}}{\sqrt{10}\times\sqrt{10}}=\frac{\sqrt{10}}{10}\)

Answer:

\(\frac{\sqrt{10}}{10}\)