Find the value of each of the six trigonometric functions of the angle theta in the figure.
theta
24
12

To find the values of the six trigonometric functions of the angle \(\theta\) in this right triangle, we can use the side lengths provided. The triangle has one leg measuring \(12\) and the other leg measuring \(24\).

### Step 1: Calculate the Hypotenuse
Using the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2} = \sqrt{12^2 + 24^2}
\]
\[
= \sqrt{144 + 576} = \sqrt{720} = 12\sqrt{5}
\]

So, the hypotenuse \(c\) is \(12\sqrt{5}\).

### Step 2: Trigonometric Functions

Now, we can find each trigonometric function based on the angle \(\theta\):

1. **Sine** \( \sin(\theta) \):
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{24}{12\sqrt{5}} = \frac{2}{\sqrt{5}} = \frac{2\sqrt{5}}{5}
   \]

2. **Cosine** \( \cos(\theta) \):
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{12}{12\sqrt{5}} = \frac{1}{\sqrt{5}} = \frac{\sqrt{5}}{5}
   \]

3. **Tangent** \( \tan(\theta) \):
   \[
   \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{24}{12} = 2
   \]

4. **Cosecant** \( \csc(\theta) \) (reciprocal of sine):
   \[
   \csc(\theta) = \frac{\text{hypotenuse}}{\text{opposite}} = \frac{12\sqrt{5}}{24} = \frac{\sqrt{5}}{2}
   \]

5. **Secant** \( \sec(\theta) \) (reciprocal of cosine):
   \[
   \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} = \frac{12\sqrt{5}}{12} = \sqrt{5}
   \]

6. **Cotangent** \( \cot(\theta) \) (reciprocal of tangent):
   \[
   \cot(\theta) = \frac{\text{adjacent}}{\text{opposite}} = \frac{12}{24} = \frac{1}{2}
   \]

### Summary of Trigonometric Function Values:
- \(\sin(\theta) = \frac{2\sqrt{5}}{5}\)
- \(\cos(\theta) = \frac{\sqrt{5}}{5}\)
- \(\tan(\theta) = 2\)
- \(\csc(\theta) = \frac{\sqrt{5}}{2}\)
- \(\sec(\theta) = \sqrt{5}\)
- \(\cot(\theta) = \frac{1}{2}\)

Would you like more details on any of these steps?

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1. How can we verify the trigonometric values using a calculator?
2. How do these trigonometric functions relate to real-world applications?
3. What is the significance of each trigonometric function in a right triangle?
4. How would these values change if the triangle's dimensions were scaled?
5. Can we use these trigonometric values to find angle \(\theta\) directly?

**Tip:** Always remember that each trigonometric function has a reciprocal pair that can help in calculations when values are known for only sine, cosine, or tangent.

Find the value of each of the six trigonometric functions of the angle theta in the figure.

Learn how to calculate the six trigonometric functions (sine, cosine, tangent, cosecant, secant, and cotangent) of an angle θ in a right triangle using side lengths. This problem involves the Pythagorean theorem and trigonometric identities. Suitable for students in Grades 9-12.

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