To find the values of the six trigonometric functions, we need to use the triangle's side lengths provided:

Given:

• Opposite side to $\theta$: 12
• Adjacent side to $\theta$: 24

First, we find the hypotenuse ($h$) using the Pythagorean theorem: $h = \sqrt{12^2 + 24^2} = \sqrt{144 + 576} = \sqrt{720} = 12\sqrt{5}$

Now, we can define each trigonometric function:

1. Sine ($\sin \theta$): $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{12\sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}$

2. Cosine ($\cos \theta$): $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{24}{12\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}$

3. Tangent ($\tan \theta$): $\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{24} = \frac{1}{2}$

4. Cosecant ($\csc \theta$): $\csc \theta = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{12\sqrt{5}}{12} = \sqrt{5}$

5. Secant ($\sec \theta$): $\sec \theta = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{12\sqrt{5}}{24} = \frac{\sqrt{5}}{2}$

6. Cotangent ($\cot \theta$): $\cot \theta = \frac{\text{adjacent}}{\text{opposite}} = \frac{24}{12} = 2$

Summary of Trigonometric Functions:

• $\sin \theta = \frac{\sqrt{5}}{5}$
• $\cos \theta = \frac{2\sqrt{5}}{5}$
• $\tan \theta = \frac{1}{2}$
• $\csc \theta = \sqrt{5}$
• $\sec \theta = \frac{\sqrt{5}}{2}$
• $\cot \theta = 2$

Would you like me to explain any of these further or clarify any details? Here are 5 questions for further exploration:

1. How do these trigonometric values relate to real-world applications?
2. What are the trigonometric functions if the triangle is scaled?
3. How can these functions be used to solve problems involving angles in a circle?
4. What is the importance of rationalizing the denominator in trigonometric expressions?
5. Can we derive these functions using unit circle definitions?

Tip: Always double-check your work by ensuring the functions satisfy known identities (e.g., $\sin^2 \theta + \cos^2 \theta = 1$).