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The diagram shows two right-angled triangles within one another, where we have one large triangle \( \triangle ABC \) and a smaller triangle inside it. The dimensions and angles given are as follows:

- \( AB = 7 \), \( AC = 3 \), and the hypotenuse \( BC \) is the longest side.
- The smaller right triangle has one side labeled \( 5 \), and the angle at \( B \) (marked as \( \theta \)) is shared by both triangles.

The problem is asking to "Calculate the value of the tangent of angle \( \theta \)" (which likely refers to \( \tan(\theta) \)).

To solve for \( \tan(\theta) \), we can use the trigonometric identity:
\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}
\]
where \( \theta \) is in the smaller right triangle. Based on the information:

- In the small triangle, the opposite side to \( \theta \) is 3, and the adjacent side (likely corresponding to \( AB \)) is 5.
- Thus, we calculate:
\[
\tan(\theta) = \frac{3}{5}
\]

Would you like a detailed step-by-step breakdown of how this is derived, or do you have any other questions?

Here are 5 related questions:
1. What is the sine of angle \( \theta \) in this triangle?
2. How would you find the length of \( BC \) using the Pythagorean theorem?
3. Can you compute the cosine of angle \( \theta \)?
4. How does knowing \( \tan(\theta) \) help in determining the slope of a line?
5. If the triangle were scaled, how would \( \tan(\theta) \) change?

**Tip:** In right-angled triangles, the tangent of an angle is often used in slope calculations and in problems involving inclines.

Calculate the value of the tangent of angle \(\theta\) in the given triangle diagram.

Learn how to calculate the tangent of angle \(\theta\) in a right triangle using trigonometric identities. This solution involves understanding trigonometric ratios such as tangent, sine, and cosine, as well as applying the Pythagorean theorem. Suitable for Grades 8-10.

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