UpStudy Free Solution:

To find the value of \(\cos A\) in the given right triangle \(\triangle ABC\), we use the definition of the cosine function in a right triangle. The cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

In \(\triangle ABC\):

- The hypotenuse \(AC = 26\)

- The adjacent side to angle \(A\) is \(BC = 10\)

- The opposite side to angle \(A\) is \(AB = 24\)

The formula for cosine is:

\[\cos A = \frac { \text { adjacent} } { \text { hypotenuse} } \]

So, we have:

\[\cos A = \frac { BC} { AC} = \frac { 10} { 26} = \frac { 5} { 13} \]

Therefore, the value of \(\cos A\) is \(\frac { 5} { 13} \).

Supplemental Knowledge

In trigonometry, the cosine of an angle in a right triangle is defined as the ratio of the length of the adjacent side to the hypotenuse. For a right triangle with angle \(A\), if \(\text { adjacent} \) is the length of the side adjacent to \(A\) and \(\text { hypotenuse} \) is the length of the hypotenuse, then:

\[\cos A = \frac { \text { adjacent} } { \text { hypotenuse} } \]

In this context, we use this definition to find \(\cos A\).

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