Let (5, -2) be a point on the terminal side of theta. Find the exact values of cos(theta), csc(theta), and tan(theta)

Given point (5, -2) lies on the terminal side of angle \theta in the Cartesian coordinate system. To find the exact values of \cos(\theta) , \csc(\theta) , and \tan(\theta) , we can use the Pythagorean identity and the definitions of trigonometric functions.

Let r be the distance from the origin to the point (5, -2) which can be calculated as follows:
r = \sqrt{5^2 + (-2)^2} = \sqrt{25 + 4} = \sqrt{29}

Now, we can find the values of \cos(\theta) , \csc(\theta) , and \tan(\theta) :
\cos(\theta) = \dfrac{x}{r} = \dfrac{5}{\sqrt{29}} = \dfrac{5\sqrt{29}}{29}

\csc(\theta) = \dfrac{r}{y} = \dfrac{\sqrt{29}}{-2} = -\dfrac{\sqrt{29}}{2}

\tan(\theta) = \dfrac{y}{x} = \dfrac{-2}{5} = -\dfrac{2}{5}

Therefore, the exact values are:
\cos(\theta) = \dfrac{5\sqrt{29}}{29}
\csc(\theta) = -\dfrac{\sqrt{29}}{2}
\tan(\theta) = -\dfrac{2}{5}

\textbf{Answer:}
\cos(\theta) = \dfrac{5\sqrt{29}}{29}, \quad \csc(\theta) = -\dfrac{\sqrt{29}}{2}, \quad \tan(\theta) = -\dfrac{2}{5}