Let (-7,-2) be a point on the terminal side of angle.

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.

Explore how to find trigonometric values for an angle with a point (-7, -2) on its terminal side. Calculate sine, cosine, tangent, cosecant, secant, and cotangent functions step-by-step using coordinate geometry and trigonometric principles.

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