Let's solve the problem given in the image.

We are asked to find the values of sine, cosine, tangent, cosecant, secant, and cotangent for the angle $\theta$ in standard position on the coordinate plane, with the point $(-7, -9)$ on its terminal side.

Step 1: Calculate the hypotenuse $r$

Using the Pythagorean theorem, we can calculate $r$, the distance from the origin to the point $(-7, -9)$. This distance is the hypotenuse of a right triangle formed by the x-coordinate and y-coordinate.

$r = \sqrt{(-7)^2 + (-9)^2} = \sqrt{49 + 81} = \sqrt{130}$ $r = \sqrt{130}$

Step 2: Calculate the trigonometric functions

Given the coordinates $(-7, -9)$, we have:

• $x = -7$
• $y = -9$
• $r = \sqrt{130}$

Now, we can find the trigonometric functions:

1. Sine: $\sin \theta = \frac{y}{r} = \frac{-9}{\sqrt{130}} = \frac{-9\sqrt{130}}{130}$

2. Cosine: $\cos \theta = \frac{x}{r} = \frac{-7}{\sqrt{130}} = \frac{-7\sqrt{130}}{130}$

3. Tangent: $\tan \theta = \frac{y}{x} = \frac{-9}{-7} = \frac{9}{7}$

4. Cosecant: $\csc \theta = \frac{r}{y} = \frac{\sqrt{130}}{-9} = \frac{-\sqrt{130}}{9}$

5. Secant: $\sec \theta = \frac{r}{x} = \frac{\sqrt{130}}{-7} = \frac{-\sqrt{130}}{7}$

6. Cotangent: $\cot \theta = \frac{x}{y} = \frac{-7}{-9} = \frac{7}{9}$

Final Answers:

$\sin \theta = \frac{-9\sqrt{130}}{130}$ $\cos \theta = \frac{-7\sqrt{130}}{130}$ $\tan \theta = \frac{9}{7}$ $\csc \theta = \frac{-\sqrt{130}}{9}$ $\sec \theta = \frac{-\sqrt{130}}{7}$ $\cot \theta = \frac{7}{9}$

Would you like further details on any of these steps or explanations?

Here are 5 related questions to expand your understanding:

1. How does the quadrant in which the terminal side lies affect the sign of trigonometric functions?
2. How can you use the unit circle to verify these trigonometric values?
3. What is the relationship between secant and cosine?
4. How does the Pythagorean theorem relate to trigonometric identities?
5. How do these trigonometric functions change if the point is in a different quadrant?

Tip: When dealing with points on the coordinate plane and finding trigonometric values, always remember to account for the signs of $x$ and $y$ depending on the quadrant the terminal side lies in.