https://r2.erweima.ai/mathbot/image/1725197975626f3kvi1nz.jpg

Um triângulo 
\[ABC\] possui lados 
\[AB = \sqrt{3} cm\], 
\[BC= 6 cm\] e 
\[CA= \sqrt{21} \space\text{cm}\].


The given triangle \(ABC\) has sides:
- \(AB = \sqrt{3} \, \text{cm}\)
- \(BC = 6 \, \text{cm}\)
- \(CA = \sqrt{21} \, \text{cm}\)

We are asked to find the measure of angle \(B\).

### Step 1: Use the Law of Cosines
The Law of Cosines relates the sides of a triangle to the cosine of one of its angles:
\[
\cos B = \frac{a^2 + c^2 - b^2}{2ac}
\]
where \(a\), \(b\), and \(c\) are the lengths of the sides opposite angles \(A\), \(B\), and \(C\), respectively. For triangle \(ABC\), the sides are:
- \(a = AC = \sqrt{21}\)
- \(b = AB = \sqrt{3}\)
- \(c = BC = 6\)

Plugging the values into the formula:
\[
\cos B = \frac{(\sqrt{21})^2 + 6^2 - (\sqrt{3})^2}{2 \times \sqrt{21} \times 6}
\]

### Step 2: Calculate
Now, let's calculate the cosine of angle \(B\).The measure of angle \(B\) in the given triangle is approximately \(10.89^\circ\).

Would you like more details on the steps, or do you have any other questions?

Here are five related questions:
1. How would you find the other angles in this triangle?
2. Can you verify the triangle's properties using the Pythagorean theorem?
3. How does the Law of Sines apply to this triangle?
4. What is the sum of all angles in a triangle, and how does it relate to this problem?
5. How would you determine if this triangle is acute, obtuse, or right-angled?

**Tip:** When solving triangles, always check if the side lengths follow the triangle inequality to ensure the triangle is valid.

Learn how to calculate angle B in triangle ABC with sides √3 cm, 6 cm, and √21 cm using the Law of Cosines. This problem involves applying trigonometric principles to determine the angle measure. Suitable for students in Grades 10-12.

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