Given the right triangle below, use the cosine function to write a trigonometric expression for the missing angle $\phi$.

• A. $\phi=\cos^{-1}\left(\frac{12}{13}\right)$
• B. $\phi=\cos^{-1}\left(\frac{13}{5}\right)$
• C. $\phi=\cos^{-1}\left(\frac{13}{12}\right)$
• D. $\phi=\cos^{-1}\left(\frac{5}{13}\right)$

Answer: (D)

Verified step by step guidance

1. Identify the sides of the right triangle: the hypotenuse is 13, the adjacent side to angle ϕ is 12, and the opposite side to angle ϕ is 5.
2. Recall the definition of the cosine function in a right triangle: \( \cos(\phi) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
3. Substitute the known values into the cosine function: \( \cos(\phi) = \frac{12}{13} \).
4. To find the angle ϕ, use the inverse cosine function: \( \phi = \cos^{-1}\left(\frac{12}{13}\right) \).
5. Compare the given options to find the correct expression for angle ϕ. The correct expression is \( \phi = \cos^{-1}\left(\frac{5}{13}\right) \), which matches the given correct answer.