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 for a right triangle: cos(ϕ) = adjacent/hypotenuse.
3. Substitute the known values into the cosine function: cos(ϕ) = 12/13.
4. To find the angle ϕ, use the inverse cosine function: ϕ = cos⁻¹(12/13).
5. Verify the expression by checking the given options and confirming that ϕ = cos⁻¹(5/13) is the correct trigonometric expression for the missing angle.