To find the remaining trigonometric functions, let's first analyze the given information:

Step 1: Understand the quadrant

1. $\cot(\theta) = \frac{24}{7}$. Since cotangent is positive, and sine is negative ($\sin(\theta) < 0$), the angle $\theta$ is in the third quadrant, where both sine and cosine are negative.

Step 2: Relation between cotangent and tangent

$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \implies \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{1}{\cot(\theta)} = \frac{7}{24}.$

Step 3: Use the Pythagorean identity to find the hypotenuse

Using the relation $\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}$, we construct a right triangle with:

• Adjacent side = 24
• Opposite side = 7

The hypotenuse $r$ is given by: $r = \sqrt{\text{adjacent}^2 + \text{opposite}^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25.$

Step 4: Determine all trigonometric functions

Now we compute the six trigonometric functions:

1. $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{-7}{25}$ (negative in the third quadrant).
2. $\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{-24}{25}$ (negative in the third quadrant).
3. $\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} = \frac{-7/25}{-24/25} = \frac{7}{24}$.
4. $\csc(\theta) = \frac{1}{\sin(\theta)} = \frac{1}{-7/25} = -\frac{25}{7}$.
5. $\sec(\theta) = \frac{1}{\cos(\theta)} = \frac{1}{-24/25} = -\frac{25}{24}$.
6. $\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} = \frac{-24/25}{-7/25} = \frac{24}{7}$.

Final Answer:

1. $\sin(\theta) = -\frac{7}{25}$
2. $\cos(\theta) = -\frac{24}{25}$
3. $\tan(\theta) = \frac{7}{24}$
4. $\csc(\theta) = -\frac{25}{7}$
5. $\sec(\theta) = -\frac{25}{24}$

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Do you want me to provide a step-by-step explanation for any specific part?

Here are 5 related questions for further exploration:

1. How do you determine the quadrant of an angle based on given trigonometric signs?
2. Can you find the angle $\theta$ in degrees or radians using inverse trigonometric functions?
3. How do the signs of trigonometric functions vary in different quadrants?
4. How can you verify the results using the Pythagorean identity?
5. What are the implications of knowing the hypotenuse when calculating all trigonometric functions?

Tip: Always remember the ASTC rule (All Students Take Calculus) to determine which trigonometric functions are positive in each quadrant.