https://file.aiquickdraw.com/mathbot/file/1726737272552oy7mbfxa.jpg

The image shows a right triangle with the following information:

- One leg of the triangle has a length of 20.
- The hypotenuse has a length of 29.
- The angle \(\theta\) is marked, and the goal is to find \(\tan(\theta)\).

The formula for \(\tan(\theta)\) is:

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

In this case, we know the length of the opposite side (20), but we need the length of the adjacent side to compute \(\tan(\theta)\). We can use the Pythagorean theorem to find the missing side. The Pythagorean theorem states:

\[
a^2 + b^2 = c^2
\]

Where:
- \(a\) and \(b\) are the legs of the triangle,
- \(c\) is the hypotenuse.

Here, we know:
- One leg: 20
- Hypotenuse: 29

Let the adjacent side be \(x\). Applying the Pythagorean theorem:

\[
x^2 + 20^2 = 29^2
\]
\[
x^2 + 400 = 841
\]
\[
x^2 = 841 - 400 = 441
\]
\[
x = \sqrt{441} = 21
\]

Now that we know the adjacent side is 21, we can calculate \(\tan(\theta)\):

\[
\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{20}{21}
\]

So, \(\tan(\theta) = \frac{20}{21}\).

Would you like further details on this, or do you have any additional questions?

Here are 5 related questions:
1. How would you calculate \(\sin(\theta)\) in this triangle?
2. How would you calculate \(\cos(\theta)\) in this triangle?
3. Can the value of \(\tan(\theta)\) ever be greater than 1? Why or why not?
4. What is the relationship between the tangent, sine, and cosine of an angle in a right triangle?
5. How would the tangent ratio change if the opposite side was longer?

**Tip:** Remember that the tangent function always involves the ratio of the opposite to adjacent sides in a right triangle, regardless of the triangle's size.

What is the value of \( \tan(\theta) \) for the given right triangle with sides 20 and 29?

Solve for the tangent of angle \( \theta \) in a right triangle with sides of 20 and 29. Using the Pythagorean theorem to find the missing side and applying the tangent formula. This problem covers trigonometry concepts suitable for Grades 8-10.

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