Question

find the six trigonometric function values of the specified angle.

cscφ = \frac{\sqrt{745}}{24}
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

secφ = \square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the hypotenuse

We know that in a right - triangle, if the opposite side \(y = 24\) and the adjacent side \(x=11\). By the Pythagorean theorem \(r=\sqrt{x^{2}+y^{2}}\), where \(r\) is the hypotenuse. So \(r=\sqrt{11^{2}+24^{2}}=\sqrt{121 + 576}=\sqrt{697}\). But we are given \(\csc\phi=\frac{r}{y}=\frac{\sqrt{745}}{24}\), so \(r = \sqrt{745}\), \(y = 24\), \(x=\sqrt{r^{2}-y^{2}}=\sqrt{745 - 576}=\sqrt{169}=13\)

Step2: Recall the formula for \(\sec\phi\)

The formula for the secant function is \(\sec\phi=\frac{r}{x}\)

Step3: Substitute the values of \(r\) and \(x\)

Since \(r=\sqrt{745}\) and \(x = 13\), then \(\sec\phi=\frac{\sqrt{745}}{13}\)

Answer:

\(\frac{\sqrt{745}}{13}\)