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The problem states that \( \tan(\theta) = \frac{12}{21} \), and we need to find all the other trigonometric functions using a right triangle.

### Step 1: Analyze the given information

We know that:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{21}
\]
So, in a right triangle, if the length of the side opposite \( \theta \) is 12, and the length of the adjacent side is 21.

### Step 2: Use the Pythagorean theorem to find the hypotenuse

The Pythagorean theorem states:
\[
\text{hypotenuse}^2 = (\text{opposite})^2 + (\text{adjacent})^2
\]

Substituting the values:
\[
\text{hypotenuse}^2 = 12^2 + 21^2 = 144 + 441 = 585
\]
\[
\text{hypotenuse} = \sqrt{585} \approx 24.19
\]

### Step 3: Find all trigonometric functions

- **Sine**:
  \[
  \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{\sqrt{585}} \approx 0.496
  \]

- **Cosine**:
  \[
  \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{21}{\sqrt{585}} \approx 0.868
  \]

- **Tangent** (already given):
  \[
  \tan(\theta) = \frac{12}{21} \approx 0.571
  \]

- **Cosecant** (reciprocal of sine):
  \[
  \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{\sqrt{585}}{12} \approx 2.016
  \]

- **Secant** (reciprocal of cosine):
  \[
  \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{\sqrt{585}}{21} \approx 1.152
  \]

- **Cotangent** (reciprocal of tangent):
  \[
  \cot(\theta) = \frac{1}{\tan(\theta)} = \frac{21}{12} = 1.75
  \]

Thus, the trigonometric functions are approximately:
- \( \sin(\theta) \approx 0.496 \)
- \( \cos(\theta) \approx 0.868 \)
- \( \tan(\theta) \approx 0.571 \)
- \( \csc(\theta) \approx 2.016 \)
- \( \sec(\theta) \approx 1.152 \)
- \( \cot(\theta) = 1.75 \)

Would you like further details or explanations? Here are 5 related questions to expand your understanding:

1. How would the trigonometric functions change if \( \theta \) were in a different quadrant?
2. What does the value of \( \tan(\theta) \) represent in terms of the angle \( \theta \)?
3. How can we use the unit circle to derive the trigonometric functions?
4. What is the significance of the Pythagorean identity in trigonometry?
5. How would you solve for trigonometric functions if given \( \sin(\theta) \) instead of \( \tan(\theta) \)?

**Tip**: Remember that the Pythagorean identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \) is always true and can help verify your trigonometric values.

Suppose that θ is an acute angle and that tan(θ) = 12/21. Use a right triangle to find all the other trigonometric functions of θ.

Learn how to calculate all trigonometric functions when tan(θ) = 12/21 using a right triangle. This problem covers key trigonometric concepts like sine, cosine, and tangent, with the Pythagorean theorem. Suitable for students in Grades 9-12.

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