In which right triangle is the value of $x$ equal to ${\cos }^{-1}\left(\frac{4.3}{6.7}\right)$?

• A. A triangle where the length of the adjacent side to angle $x$ is $6.7$ and the hypotenuse is $4.3$
• B. A triangle where the length of the opposite side to angle $x$ is $4.3$ and the hypotenuse is $6.7$
• C. A triangle where the length of the opposite side to angle $x$ is $6.7$ and the adjacent side is $4.3$
• D. A triangle where the length of the adjacent side to angle $x$ is $4.3$ and the hypotenuse is $6.7$

Answer: (D)

Verified step by step guidance

1. Recall the definition of cosine in a right triangle: \(\cos(x) = \frac{\text{adjacent side}}{\text{hypotenuse}}\).
2. The expression \(x = \cos^{-1}\left(\frac{4.3}{6.7}\right)\) means that the cosine of angle \(x\) is \(\frac{4.3}{6.7}\).
3. Identify which sides correspond to the numerator and denominator in the fraction \(\frac{4.3}{6.7}\): the numerator (4.3) represents the length of the adjacent side to angle \(x\), and the denominator (6.7) represents the hypotenuse.
4. Compare this ratio to the given triangle options to find the one where the adjacent side to angle \(x\) is 4.3 and the hypotenuse is 6.7.
5. Confirm that this triangle matches the cosine ratio, which means \(x\) is the angle whose cosine is \(\frac{4.3}{6.7}\), consistent with the inverse cosine expression.