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.