To determine the correct values for [tex]\(\cos A\)[/tex] and [tex]\(\tan A\)[/tex] given that [tex]\(\sin A = \frac{7}{25}\)[/tex], we will proceed step-by-step, applying trigonometric identities.
### Step 1: Calculate [tex]\(\cos A\)[/tex]
Given [tex]\(\sin A = \frac{7}{25}\)[/tex], we can use the Pythagorean identity:
[tex]\[
\sin^2 A + \cos^2 A = 1
\][/tex]
First, we calculate [tex]\(\sin^2 A\)[/tex]:
[tex]\[
\sin^2 A = \left(\frac{7}{25}\right)^2 = \frac{49}{625}
\][/tex]
Next, we use the identity to find [tex]\(\cos^2 A\)[/tex]:
[tex]\[
\cos^2 A = 1 - \sin^2 A = 1 - \frac{49}{625}
\][/tex]
[tex]\[
\cos^2 A = \frac{625}{625} - \frac{49}{625} = \frac{576}{625}
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
\cos A = \sqrt{\frac{576}{625}} = \frac{24}{25}
\][/tex]
### Step 2: Calculate [tex]\(\tan A\)[/tex]
Using the definition of [tex]\(\tan A\)[/tex]:
[tex]\[
\tan A = \frac{\sin A}{\cos A}
\][/tex]
Substituting the given values:
[tex]\[
\tan A = \frac{\frac{7}{25}}{\frac{24}{25}} = \frac{7}{24}
\][/tex]
### Step 3: Identify the Correct Option
We need to match the calculated values of [tex]\(\cos A\)[/tex] and [tex]\(\tan A\)[/tex] to the given options:
a. [tex]\(\cos A = \frac{24}{25}\)[/tex] and [tex]\(\tan A = \frac{7}{24}\)[/tex]
c. [tex]\(\cos A = \frac{24}{25}\)[/tex] and [tex]\(\tan A = \frac{7}{25}\)[/tex]
b. [tex]\(\cos A = \frac{7}{25}\)[/tex] and [tex]\(\tan A = \frac{24}{25}\)[/tex]
d. [tex]\(\cos A = \frac{24}{7}\)[/tex] and [tex]\(\tan A = \frac{7}{24}\)[/tex]
From our calculations, we found:
[tex]\[
\cos A = \frac{24}{25}, \quad \tan A = \frac{7}{24}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{\text{a. } \cos A = \frac{24}{25}, \tan A = \frac{7}{24}}
\][/tex]
