Question

suppose abc is a right triangle with sides of lengths a, b, and c and right angle at c. find the unknown side length using the pythagorean theorem and then find the value of the indicated trigonometric function of the given angle. rationalize the denominator if applicable. find sec (b) when a = 5 and b = 6. a. \\(\frac{\sqrt{61}}{5}\\) b. \\(\frac{5\sqrt{61}}{6}\\) c. \\(\frac{6\sqrt{61}}{61}\\) d. \\(\frac{5\sqrt{61}}{61}\\)

Explanation:

Step1: Find the hypotenuse \(c\)

By the Pythagorean theorem \(c^{2}=a^{2}+b^{2}\). Given \(a = 5\) and \(b=6\), then \(c=\sqrt{a^{2}+b^{2}}=\sqrt{5^{2}+6^{2}}=\sqrt{25 + 36}=\sqrt{61}\).

Step2: Recall the definition of secant

The secant function is defined as \(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}\). For angle \(B\), the adjacent side to \(B\) is \(a = 5\) and the hypotenuse is \(c=\sqrt{61}\). So \(\sec(B)=\frac{c}{a}=\frac{\sqrt{61}}{5}\).

Answer:

A. \(\frac{\sqrt{61}}{5}\)