(12) Use an appropriate Pythagorean identity to find the indicated value for an acute angle \( \theta \). Write your answer as a simplifiec denominators if necessary. Given \( \tan \theta=\frac{33}{56} \), find the exact value of \( \sec \theta \). \[ \sec \theta=\square 18 \]

To find \( \sec \theta \) from \( \tan \theta = \frac{33}{56} \), first recall the relationship between tangent and secant: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \). This implies that we can think of a right triangle where the opposite side is 33 and the adjacent side is 56. Using the Pythagorean theorem, the hypotenuse \( r \) can be calculated as follows: \[ r = \sqrt{33^2 + 56^2} = \sqrt{1089 + 3136} = \sqrt{4225} = 65. \] Now that we have the hypotenuse, we can find \( \sec \theta \), which is \( \frac{r}{\text{adjacent}} = \frac{65}{56} \). Thus, \[ \sec \theta = \frac{65}{56}. \]