1. **Problem Statement:** Given that $\tan \theta = \frac{7}{24}$, find $\sec \theta$. 2. **Recall the definitions and formulas:** - $\tan \theta = \frac{\sin \theta}{\cos \theta}$. - $\sec \theta = \frac{1}{\cos \theta}$. - Using the Pythagorean identity: $1 + \tan^2 \theta = \sec^2 \theta$. 3. **Calculate $\sec \theta$ using the identity:** $$\sec^2 \theta = 1 + \tan^2 \theta = 1 + \left(\frac{7}{24}\right)^2 = 1 + \frac{49}{576} = \frac{576}{576} + \frac{49}{576} = \frac{625}{576}.$$ 4. **Take the square root to find $\sec \theta$:** $$\sec \theta = \pm \sqrt{\frac{625}{576}} = \pm \frac{25}{24}.$$ 5. **Determine the sign of $\sec \theta$:** Since $\tan \theta = \frac{7}{24}$ is positive, $\theta$ is in the first or third quadrant. - In the first quadrant, $\cos \theta > 0$ so $\sec \theta > 0$. - In the third quadrant, $\cos \theta < 0$ so $\sec \theta < 0$. Without additional information, the principal value is positive. **Final answer:** $$\boxed{\sec \theta = \frac{25}{24}}.$$