find the exact value of the expression. sec (tan -1 3/8)

Explanation:

Step1: Let $\theta=\tan^{-1}\frac{3}{8}$

This means $\tan\theta = \frac{3}{8}$, and $\theta$ is an angle in a right - triangle where the opposite side $y = 3$ and the adjacent side $x = 8$.

Step2: Use the Pythagorean theorem to find the hypotenuse $r$

By the Pythagorean theorem $r=\sqrt{x^{2}+y^{2}}$. Substituting $x = 8$ and $y = 3$, we get $r=\sqrt{8^{2}+3^{2}}=\sqrt{64 + 9}=\sqrt{73}$.

Step3: Recall the definition of secant

We know that $\sec\theta=\frac{r}{x}$. Since $r=\sqrt{73}$ and $x = 8$, then $\sec\theta=\frac{\sqrt{73}}{8}$. And since $\theta=\tan^{-1}\frac{3}{8}$, $\sec(\tan^{-1}\frac{3}{8})=\frac{\sqrt{73}}{8}$.

Answer:

$\frac{\sqrt{73}}{8}$