Question

find the six trigonometric function values of the specified angle.
secφ = \frac{\sqrt{745}}{13}
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)
cotφ = \square
(simplify your answer, including any radicals. use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Find the adjacent side, opposite side and hypotenuse

We know that \(\sec\phi=\frac{\text{hypotenuse}}{\text{adjacent}}=\frac{\sqrt{745}}{13}\), so the adjacent side \(x = 13\), hypotenuse \(r=\sqrt{745}\).
By the Pythagorean theorem \(r^{2}=x^{2}+y^{2}\), then \(y=\sqrt{r^{2}-x^{2}}=\sqrt{745 - 169}=\sqrt{576} = 24\) (since \(y>0\) in the context of the right - triangle trigonometry for the non - right angle \(\phi\)).

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

The formula for \(\cot\phi=\frac{\text{adjacent}}{\text{opposite}}\)

Step3: Substitute the values of adjacent and opposite

Substitute \(x = 13\) and \(y = 24\) into the formula for \(\cot\phi\), we get \(\cot\phi=\frac{13}{24}\)

Answer:

\(\frac{13}{24}\)