Question

find the secant of \\( \angle r \\).

write your answer in simplified, rationalized form. do not round.

\\( \sec ( r ) = \\)

Explanation:

Step1: Recall the definition of secant

In a right - triangle, \(\sec\theta=\frac{\text{hypotenuse}}{\text{adjacent}}\). For \(\angle R\), the hypotenuse is \(QR = \sqrt{15}\), and we need to find the adjacent side.

Step2: Identify the adjacent side to \(\angle R\)

Using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\) (where \(c = QR=\sqrt{15}\), \(b = RS=\sqrt{6}\), and \(a\) is the side adjacent to \(\angle R\)). But we can also directly use the ratio. The adjacent side to \(\angle R\) is \(RS=\sqrt{6}\) (since \(\sec R=\frac{QR}{RS}\)).

Step3: Calculate \(\sec R\)

\(\sec R=\frac{QR}{RS}=\frac{\sqrt{15}}{\sqrt{6}}\). Rationalize the denominator: \(\frac{\sqrt{15}}{\sqrt{6}}\times\frac{\sqrt{6}}{\sqrt{6}}=\frac{\sqrt{90}}{6}=\frac{3\sqrt{10}}{6}=\frac{\sqrt{10}}{2}\)

Answer:

\(\frac{\sqrt{10}}{2}\)