Given the point $(-7, -2)$ on the terminal side of an angle $\theta$, we can find several trigonometric values for $\theta$. Here's the step-by-step process:

1. Calculate the radius $r$:

   The radius $r$ (also known as the hypotenuse in the right triangle formed by the point $(-7, -2)$) is calculated using the Pythagorean theorem:

   $r = \sqrt{x^2 + y^2} = \sqrt{(-7)^2 + (-2)^2} = \sqrt{49 + 4} = \sqrt{53}$

2. Find the trigonometric functions:

   Using the coordinates $(-7, -2)$ and the radius $r = \sqrt{53}$, we can find the six trigonometric functions.

   • Sine ($\sin \theta$): $\sin \theta = \frac{y}{r} = \frac{-2}{\sqrt{53}} = -\frac{2}{\sqrt{53}}$

   • Cosine ($\cos \theta$): $\cos \theta = \frac{x}{r} = \frac{-7}{\sqrt{53}} = -\frac{7}{\sqrt{53}}$

   • Tangent ($\tan \theta$): $\tan \theta = \frac{y}{x} = \frac{-2}{-7} = \frac{2}{7}$

   • Cosecant ($\csc \theta$): $\csc \theta = \frac{r}{y} = \frac{\sqrt{53}}{-2} = -\frac{\sqrt{53}}{2}$

   • Secant ($\sec \theta$): $\sec \theta = \frac{r}{x} = \frac{\sqrt{53}}{-7} = -\frac{\sqrt{53}}{7}$

   • Cotangent ($\cot \theta$): $\cot \theta = \frac{x}{y} = \frac{-7}{-2} = \frac{7}{2}$

So, the trigonometric values for the angle $\theta$ are:

$\sin \theta = -\frac{2}{\sqrt{53}}, \quad \cos \theta = -\frac{7}{\sqrt{53}}, \quad \tan \theta = \frac{2}{7}, \quad \csc \theta = -\frac{\sqrt{53}}{2}, \quad \sec \theta = -\frac{\sqrt{53}}{7}, \quad \cot \theta = \frac{7}{2}$

Would you like more details or have any questions about these calculations?

Here are some related questions for further practice:

1. Find the trigonometric values for the point $(3, -4)$ on the terminal side of an angle.
2. Calculate the trigonometric functions for the point $(-5, 12)$.
3. Determine the trigonometric values for the point $(8, -15)$.
4. What are the trigonometric functions for the point $(-9, 40)$?
5. Compute the trigonometric values for the point $(6, -8)$.
6. Find the trigonometric functions for the point $(-10, 24)$.
7. Determine the trigonometric values for the point $(7, -24)$.
8. What are the trigonometric functions for the point $(-1, 1)$?

Tip: When dealing with trigonometric functions of angles in different quadrants, remember the ASTC rule (All Students Take Calculus) to determine the signs of the functions.