Given a right triangle where the length of the adjacent side to angle $x$ is $1$ and the hypotenuse is $2$, in which triangle is the value of $x$ equal to ${cos}^{-1}\left(1,/,2\right)$ (that is, $x=arccos\left(\frac{1}{2}\right)$)?

• A. A triangle with an angle $x$ where the opposite side is $1$ and the hypotenuse is $2$
• B. A triangle with an angle $x$ where the opposite side is $2$ and the hypotenuse is $1$
• C. A triangle with an angle $x$ where the adjacent side is $2$ and the hypotenuse is $1$
• D. A triangle with an angle $x$ where the adjacent side is $1$ and the hypotenuse is $2$

Answer: (D)

Verified step by step guidance

1. Recall the definition of cosine in a right triangle: for an angle \(x\), \(\cos(x) = \frac{\text{adjacent side}}{\text{hypotenuse}}\).
2. Given that \(x = \arccos\left(\frac{1}{2}\right)\), this means \(\cos(x) = \frac{1}{2}\).
3. Identify the triangle where the ratio of the adjacent side to the hypotenuse equals \(\frac{1}{2}\), which matches the given adjacent side length of 1 and hypotenuse length of 2.
4. Check the other options by calculating their cosine ratios: for example, if the opposite side is 1 and hypotenuse is 2, \(\cos(x) = \frac{\text{adjacent}}{2}\), which is not necessarily \(\frac{1}{2}\) without knowing the adjacent side; similarly, if the hypotenuse is smaller than the side, the ratio would be invalid.
5. Conclude that the triangle with adjacent side 1 and hypotenuse 2 corresponds to \(x = \arccos\left(\frac{1}{2}\right)\) because it satisfies the cosine ratio definition.