Find the value of each of the six trigonometric functions of the angle theta in the figure.
theta
24
12
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Part 1
sine theta equals
  
enter your response here
​(Simplify your answer. Use integers or fractions for any numbers in the expression. Type an exact​ answer, using radicals as​ needed.)


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., \(\sin^2 \theta + \cos^2 \theta = 1\)).

Find the value of each of the six trigonometric functions of the angle theta in the figure. Given: Opposite side to θ = 12, Adjacent side to θ = 24.

Find the exact values of all six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) for an angle θ in a right triangle with opposite side 12 and adjacent side 24. This problem uses the Pythagorean theorem and fundamental trigonometric identities. Ideal for students in Grades 9-11.

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