We are given the point $(-7, 24)$ on the terminal side of an angle $\theta$. To find the six trigonometric functions, we first determine the hypotenuse $r$:

Step 1: Find $r$

$r = \sqrt{x^2 + y^2}$ $r = \sqrt{(-7)^2 + 24^2}$ $r = \sqrt{49 + 576}$ $r = \sqrt{625} = 25$

Step 2: Compute the Six Trigonometric Functions

1. Sine ($\sin \theta$): $\sin \theta = \frac{y}{r} = \frac{24}{25}$

2. Cosine ($\cos \theta$): $\cos \theta = \frac{x}{r} = \frac{-7}{25}$

3. Tangent ($\tan \theta$): $\tan \theta = \frac{y}{x} = \frac{24}{-7} = -\frac{24}{7}$

4. Cosecant ($\csc \theta$) (Reciprocal of sine): $\csc \theta = \frac{1}{\sin \theta} = \frac{25}{24}$

5. Secant ($\sec \theta$) (Reciprocal of cosine): $\sec \theta = \frac{1}{\cos \theta} = \frac{25}{-7} = -\frac{25}{7}$

6. Cotangent ($\cot \theta$) (Reciprocal of tangent): $\cot \theta = \frac{1}{\tan \theta} = \frac{-7}{24}$

Final Answer:

$\sin \theta = \frac{24}{25}, \quad \cos \theta = -\frac{7}{25}, \quad \tan \theta = -\frac{24}{7}$ $\csc \theta = \frac{25}{24}, \quad \sec \theta = -\frac{25}{7}, \quad \cot \theta = -\frac{7}{24}$

Would you like a more detailed explanation of any step?

Related Questions:

1. How do you determine the quadrant of an angle given a point on its terminal side?
2. What is the significance of the hypotenuse in trigonometry?
3. How can you use trigonometric identities to verify these values?
4. How do you find the reference angle for $\theta$ given this point?
5. What happens to these trigonometric functions if the point were in a different quadrant?

Tip: Remember that signs of trigonometric functions depend on the quadrant!