https://file.aiquickdraw.com/mathbot/file/1728385798520c27nzj11.jpg

The image presents a right triangle, and you are asked to calculate \( \tan(\beta) \) and \( \tan(\alpha) \), where \( \beta \) and \( \alpha \) are two angles in the triangle.

### Given information:
- The opposite side to \( \beta \) is labeled as 29.
- The adjacent side to \( \beta \) (which is opposite to \( \alpha \)) is labeled as \( 29 + 1 = 30 \).
- The hypotenuse is not directly labeled, but it can be calculated using the Pythagorean theorem.

### Step-by-step solution:

1. **Calculate \( \tan(\beta) \)**:
   \[
   \tan(\beta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{29}{30}
   \]

2. **Calculate \( \tan(\alpha) \)**:
   \[
   \tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}} = \frac{30}{29}
   \]

So, the values of the tangents are:
- \( \tan(\beta) = \frac{29}{30} \)
- \( \tan(\alpha) = \frac{30}{29} \)

Would you like further clarification on these calculations, or do you have any other questions?

Here are some related questions:
1. How do we use the Pythagorean theorem to find the hypotenuse in this triangle?
2. What is the relationship between \( \tan(\alpha) \) and \( \tan(\beta) \) in this triangle?
3. Can we find the exact values of \( \alpha \) and \( \beta \) using the inverse tangent function?
4. How would we calculate the sine and cosine of \( \alpha \) and \( \beta \)?
5. How can we generalize this approach to other right triangles?

**Tip**: The tangent of an angle in a right triangle is always the ratio of the opposite side to the adjacent side for that specific angle.

Calculate tan(β) and tan(α) based on the right triangle in the image, where one side is labeled 29, the other side is 29 + 1, and the angles β and α are indicated.

This problem involves calculating tan(β) and tan(α) in a right triangle. The sides are labeled as 29 and 30, and the solution uses basic trigonometry to find the tangent of the angles. Ideal for students in Grades 9-12 learning about trigonometric ratios.

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