https://r2.erweima.ai/mathbot/image/17264979919078mrcxkfc.jpg

The given problem asks to determine the length of side \( BC \) in triangle \( ABC \) where:

- \( AB = 4 \)
- \( AC = 4\sqrt{7} \)
- \( \angle B = \frac{2\pi}{3} \) (which is equivalent to \( 120^\circ \))

We can solve this using the **Law of Cosines**, which states:

\[
BC^2 = AB^2 + AC^2 - 2 \cdot AB \cdot AC \cdot \cos(\angle B)
\]

### Step-by-step solution:

1. **Substitute the known values:**

\[
BC^2 = 4^2 + (4\sqrt{7})^2 - 2 \cdot 4 \cdot 4\sqrt{7} \cdot \cos\left(120^\circ\right)
\]

2. **Simplify the terms:**

\[
BC^2 = 16 + 16 \cdot 7 - 2 \cdot 4 \cdot 4\sqrt{7} \cdot \left(-\frac{1}{2}\right)
\]

\[
BC^2 = 16 + 112 + 64\sqrt{7} \cdot \frac{1}{2}
\]

\[
BC^2 = 128 + 32\sqrt{7}
\]

3. **Find the square root to get \( BC \):**

Thus, \( BC = \sqrt{128 + 32\sqrt{7}} \).

This gives the length of \( BC \). 

Would you like further clarification or additional steps on this? Below are some questions to expand your understanding:

1. How is the Law of Cosines derived?
2. What are some common situations where the Law of Cosines is useful?
3. How can you convert angles from radians to degrees?
4. Why is the cosine of \(120^\circ\) equal to \( -\frac{1}{2} \)?
5. What are other ways to calculate the third side of a triangle?

**Tip**: Understanding the Law of Cosines is crucial when solving non-right triangles.

Determinați lungimea laturii BC în triunghiul ABC cu AB = 4, AC = 4√7 și unghiul B = 120° (2π/3).

Learn how to find the length of side BC in triangle ABC with sides AB = 4, AC = 4√7, and angle B = 120°. This problem applies the Law of Cosines, a key concept in trigonometry for solving non-right triangles.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation