Find the value of each of the six trigonometric functions of the angle theta in the figure.
theta
12
6
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Part 1
sine theta equals

StartFraction StartRoot 3 EndRoot Over 2 EndFraction
​(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 of an angle \(\theta\), we usually need the information about a right triangle or a unit circle where the angle is located. However, you provided a numerical clue ("12" and "6"), but the exact context of these numbers isn't clear (whether they are lengths of sides in a right triangle, coordinates of a point on the unit circle, etc.).

Let’s assume this is a right triangle with sides labeled relative to \(\theta\), and "12" and "6" are the lengths of two sides of the triangle. Given a standard right triangle, we have three sides: opposite, adjacent, and hypotenuse. Since the context suggests finding \(\sin(\theta)\), we'll explore this possibility:

### Step-by-Step Solution

1. **Assume the side lengths:**
   Let:
   - Opposite side = \(6\)
   - Adjacent side = \(12\)

2. **Find the hypotenuse:**
   Using the Pythagorean theorem:

   \[
   \text{Hypotenuse} = \sqrt{(6^2 + 12^2)} = \sqrt{36 + 144} = \sqrt{180} = 6\sqrt{5}
   \]

3. **Find the six trigonometric functions:**

   - **Sine:**

     \[
     \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{6}{6\sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}
     \]

   - **Cosine:**

     \[
     \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{6\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
     \]

   - **Tangent:**

     \[
     \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{6}{12} = \frac{1}{2}
     \]

   - **Cosecant:**

     \[
     \csc(\theta) = \frac{1}{\sin(\theta)} = \sqrt{5}
     \]

   - **Secant:**

     \[
     \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{5}}{2}
     \]

   - **Cotangent:**

     \[
     \cot(\theta) = \frac{1}{\tan(\theta)} = 2
     \]

If the values (12 and 6) are different, please clarify the setup or provide a diagram. 

Would you like further details or clarifications on these results?

### Related Questions to Explore:

1. What are the trigonometric values if the triangle is isosceles?
2. How does the unit circle relate to these trigonometric functions?
3. What are the reciprocal trigonometric functions and their significance?
4. How would these values change if the triangle is in a different quadrant?
5. Can you calculate the exact angle in degrees or radians from the sine value?

### Tip:
Always double-check if the given side lengths are for a right triangle and determine which sides correspond to the angle \(\theta\).

Learn how to find the values of sine, cosine, tangent, cosecant, secant, and cotangent of angle theta using given side lengths in a right triangle. This problem involves applying the Pythagorean theorem and trigonometric ratios to determine exact trigonometric values. Suitable for high school students in Grades 10-12.

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