Triangle ABC has a right angle at C. The length of side BC is 2.55 cm,
and the length of side AC is 5.89 cm. Sketch the triangle and find
∠ABC, giving your answer correct to the nearest degree.

To solve this problem, we will use trigonometric ratios to find the angle ∠ABC. Since triangle ABC is a right triangle with a right angle at C, we can apply the sine, cosine, or tangent functions.

Given:
- \( BC = 2.55 \) cm (opposite side of ∠ABC)
- \( AC = 5.89 \) cm (adjacent side to ∠ABC)

We can use the tangent function because tangent of an angle in a right triangle is the ratio of the length of the opposite side to the adjacent side.

\[
\tan(\angle ABC) = \frac{BC}{AC} = \frac{2.55}{5.89}
\]

Let's calculate this ratio:

\[
\tan(\angle ABC) = \frac{2.55}{5.89} \approx 0.4324
\]

Next, we find the angle using the inverse tangent (arctan) function:

\[
\angle ABC = \tan^{-1}(0.4324)
\]

Now, let's calculate the angle.

\[
\angle ABC \approx \tan^{-1}(0.4324) \approx 23.4^\circ
\]

Rounding to the nearest degree:

\[
\angle ABC \approx 23^\circ
\]

### Summary:
- ∠ABC is approximately 23°.

Would you like a detailed sketch of the triangle or any further explanation?

Here are some related questions you might find interesting:
1. What is the length of side AB using the Pythagorean theorem?
2. How would the value of ∠ABC change if side BC were longer?
3. What is the sine of ∠ABC?
4. What is the cosine of ∠ABC?
5. How would you find ∠ACB in this triangle?

**Tip:** When dealing with right triangles, always remember the basic trigonometric ratios: sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent).

Learn how to find angle ABC in a right triangle with a given side length of BC = 2.55 cm and AC = 5.89 cm. Using trigonometric ratios and the inverse tangent function, calculate ∠ABC accurately to the nearest degree. Suitable for high school students learning trigonometry.

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