https://file.aiquickdraw.com/mathbot/file/172174737971085ksynjr.png

To solve questions 2 and 3, let's proceed as follows:

### 2. Calculate the size of angle B

In a right-angled triangle, the two non-right angles are complementary. This means their sum is 90°.

Given:
- \( \tan(A) = \frac{3}{2} \)
- The triangle is right-angled at C.

Let's find angle A first:
\[ \tan(A) = \frac{3}{2} \]
\[ A = \tan^{-1}\left(\frac{3}{2}\right) \]

Using a calculator to find the inverse tangent:
\[ A \approx 56.31^\circ \]

Now, since the sum of angles in a triangle is 180° and one angle is 90° (right angle):
\[ B = 90^\circ - A \]
\[ B = 90^\circ - 56.31^\circ \]
\[ B \approx 33.69^\circ \]

### 3. Determine the length of AB

Using the Pythagorean theorem:
\[ AB = \sqrt{AC^2 + BC^2} \]

Given:
- \( AC = 4 \) units
- \( BC = 6 \) units

\[ AB = \sqrt{4^2 + 6^2} \]
\[ AB = \sqrt{16 + 36} \]
\[ AB = \sqrt{52} \]
\[ AB = 2\sqrt{13} \approx 7.21 \text{ units} \]

### Summary

2. The size of angle B is approximately \( 33.69^\circ \).
3. The length of AB is \( 2\sqrt{13} \approx 7.21 \) units.

Do you want details or have any questions?

Here are 5 more questions you can ask:

1. How do you derive the trigonometric ratios for a right-angled triangle?
2. What are complementary and supplementary angles?
3. How is the Pythagorean theorem applied in various scenarios?
4. Can you explain the unit circle and how it relates to trigonometric functions?
5. How can trigonometric identities simplify solving complex equations?

**Tip:** When dealing with right-angled triangles, always check if you can use trigonometric ratios or the Pythagorean theorem to find missing sides or angles.

Learn how to calculate angle B and length AB in a right-angled triangle using trigonometric ratios and the Pythagorean theorem. Detailed step-by-step solution provided. Suitable for high school students.

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