https://file.aiquickdraw.com/mathbot/file/1729557468680mdheh1rp.jpg

The given image depicts a triangle with:

- Side \( AC = 4 \)
- Side \( AB = 7 \)
- Angle \( \angle ACB = 35^\circ \)

We need to find the length of side \( b \), which is the side \( BC \).

### Step-by-Step Solution Using the Law of Cosines:

The Law of Cosines is used when two sides and the included angle are known. It is given by:

\[
b^2 = a^2 + c^2 - 2ac \cdot \cos(\theta)
\]

Where:
- \( a = 4 \) (side AC)
- \( c = 7 \) (side AB)
- \( \theta = 35^\circ \) (angle \( \angle ACB \))

Substitute the known values:

\[
b^2 = 4^2 + 7^2 - 2(4)(7) \cdot \cos(35^\circ)
\]
\[
b^2 = 16 + 49 - 56 \cdot \cos(35^\circ)
\]
\[
b^2 = 65 - 56 \cdot \cos(35^\circ)
\]

Now, calculate \( \cos(35^\circ) \approx 0.8192 \):

\[
b^2 = 65 - 56 \cdot 0.8192
\]
\[
b^2 = 65 - 45.8752
\]
\[
b^2 = 19.1248
\]

Finally, take the square root:

\[
b = \sqrt{19.1248} \approx 4.374
\]

Thus, the length of side \( b \) (side \( BC \)) is approximately **4.37 units**.

---

Would you like more details or explanations on this process? 

### Related Questions:
1. How do you calculate angles in a triangle using the Law of Sines?
2. What is the Law of Cosines used for in general geometry?
3. How can we find the area of this triangle using the given data?
4. Can the Law of Sines be applied here, and how would it change the approach?
5. What are the steps to solve for the third angle in this triangle?

### Tip:
Always ensure that your calculator is in degree mode when working with trigonometric functions involving degrees, such as cosine of 35°.

Find the length of side b in a triangle with sides AC = 4, AB = 7, and angle ∠ACB = 35°.

Learn how to calculate the length of side BC using the Law of Cosines for a triangle where sides AC = 4, AB = 7, and angle ACB = 35°. This problem covers key trigonometric concepts suitable for students in Grade 10-11.

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