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To find the exact values of the remaining five trigonometric functions based on \( \tan(\theta) = \frac{1}{7} \), we will use trigonometric identities and the Pythagorean theorem.

### Step 1: Define the sides of the triangle
Since \( \tan(\theta) = \frac{1}{7} \), we can interpret this as the opposite side being 1 and the adjacent side being 7 in a right triangle.

### Step 2: Find the hypotenuse
We use the Pythagorean theorem to find the hypotenuse \( r \):
\[
r = \sqrt{1^2 + 7^2} = \sqrt{1 + 49} = \sqrt{50} = 5\sqrt{2}
\]

### Step 3: Calculate the remaining five trigonometric functions

1. **Sine** \( \sin(\theta) \):
   \[
   \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{1}{5\sqrt{2}} = \frac{\sqrt{2}}{10}
   \]

2. **Cosine** \( \cos(\theta) \):
   \[
   \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{7}{5\sqrt{2}} = \frac{7\sqrt{2}}{10}
   \]

3. **Cosecant** \( \csc(\theta) \):
   Cosecant is the reciprocal of sine:
   \[
   \csc(\theta) = \frac{1}{\sin(\theta)} = \frac{10}{\sqrt{2}} = 5\sqrt{2}
   \]

4. **Secant** \( \sec(\theta) \):
   Secant is the reciprocal of cosine:
   \[
   \sec(\theta) = \frac{1}{\cos(\theta)} = \frac{10}{7\sqrt{2}} = \frac{5\sqrt{2}}{7}
   \]

5. **Cotangent** \( \cot(\theta) \):
   Cotangent is the reciprocal of tangent:
   \[
   \cot(\theta) = \frac{1}{\tan(\theta)} = 7
   \]

Thus, the six trigonometric functions are:
- \( \sin(\theta) = \frac{\sqrt{2}}{10} \)
- \( \cos(\theta) = \frac{7\sqrt{2}}{10} \)
- \( \tan(\theta) = \frac{1}{7} \) (given)
- \( \csc(\theta) = 5\sqrt{2} \)
- \( \sec(\theta) = \frac{5\sqrt{2}}{7} \)
- \( \cot(\theta) = 7 \)

Let me know if you need any details or have questions!

Here are five related questions to deepen your understanding:
1. How would the trigonometric functions change if \( \tan(\theta) \) was negative?
2. What is the geometric interpretation of \( \sin(\theta) \) and \( \cos(\theta) \) on the unit circle?
3. How do you find the trigonometric functions for angles in different quadrants?
4. Can you derive the Pythagorean identity using the values of \( \sin(\theta) \) and \( \cos(\theta) \)?
5. How does the concept of reference angles apply to trigonometric functions?

**Tip:** The signs of the trigonometric functions change depending on the quadrant in which the angle lies, so always consider that when dealing with non-acute angles.

Use the definition or identities to find the exact value of each of the remaining five trigonometric functions of the acute angle θ, given tan(θ) = 1/7.

Learn how to calculate the exact values of the remaining five trigonometric functions for the acute angle θ, given tan(θ) = 1/7. The solution involves using the Pythagorean theorem and trigonometric identities to find sin(θ), cos(θ), csc(θ), sec(θ), and cot(θ). Suitable for high school students in Grades 9-12.

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