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.