https://r2.erweima.ai/mathbot/image/1726004996716l2dz83h5.jpg

To solve for \(\tan(\beta)\) in this triangle, we use the definition of tangent in a right triangle, which is:

\[
\tan(\beta) = \frac{\text{opposite}}{\text{adjacent}}
\]

From the diagram:
- The side opposite to angle \(\beta\) is \(4\) (side \(AC\)).
- The side adjacent to angle \(\beta\) is \(3\) (side \(BC\)).

Thus, we can calculate:

\[
\tan(\beta) = \frac{4}{3}
\]

So, \(\tan(\beta) = \frac{4}{3}\) or approximately \(1.333\).

Would you like more details or explanations on this? Here are 5 related questions to consider:
1. How do we determine the sine and cosine of angle \(\beta\) in this triangle?
2. Can the Pythagorean theorem help verify the side lengths of this triangle?
3. How does the tangent function relate to other trigonometric ratios like sine and cosine?
4. How would we calculate the value of \(\beta\) in degrees or radians?
5. How does changing the length of the sides affect the value of \(\tan(\beta)\)?

**Tip:** Remember, in any right triangle, tangent, sine, and cosine are all ratios of side lengths, so they're easy to compute if the sides are known!

Learn how to calculate the tangent of angle β in a right triangle using the formula tan(β) = opposite/adjacent. In this triangle, the sides opposite and adjacent to β are 4 and 3, respectively. This problem is suitable for students in Grades 8-10.

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