https://file.aiquickdraw.com/mathbot/file/17261797454308ufe0i8k.jpg

The image shows a right triangle \( \triangle ABC \), with the following known values:

- \( AB = 5 \) (side opposite the right angle),
- \( AC = 12 \) (one leg),
- \( BC = 8 \) (other leg).

We are tasked with finding the angle \( \angle ACB \), labeled as \( x \).

### Solution

To find the angle \( x \), we can use trigonometry. Since we know the adjacent side \( BC = 8 \) and the opposite side \( AB = 5 \), we can apply the tangent function:

\[
\tan(x) = \frac{\text{opposite}}{\text{adjacent}} = \frac{AB}{BC} = \frac{5}{8}
\]

Now we can calculate the angle \( x \) by taking the arctangent (inverse tangent) of \( \frac{5}{8} \):

\[
x = \tan^{-1}\left(\frac{5}{8}\right)
\]

Let me calculate the exact value of \( x \).The angle \( x \) is approximately \( 32.01^\circ \).

Would you like more details on the calculation or explanation?

Here are 5 related questions to deepen your understanding:
1. How do we calculate an angle using sine or cosine instead of tangent?
2. What are some other real-life applications of trigonometric ratios in right triangles?
3. How can we use the Pythagorean theorem to verify the side lengths of a right triangle?
4. How does the inverse trigonometric function differ from the regular trigonometric function?
5. What is the significance of rounding in trigonometric calculations?

**Tip:** Always double-check which sides are opposite, adjacent, or the hypotenuse relative to the angle in question when using trigonometric ratios.

Find the indicated part of triangle ABC, where AB = 5, BC = 8, and AC = 12, and angle x is opposite to side AB.

Learn how to find the angle in a right triangle ABC where AB = 5, BC = 8, and AC = 12 using the tangent function. The problem explores key trigonometric concepts and is suitable for students in Grades 8-10.

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