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To find the value of \( x \), we can use the **Law of Cosines**. The Law of Cosines is used when we have two sides and the included angle, or when we need to find the third side of a triangle.

The formula is given by:

\[
x^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]

Where:
- \( x \) is the side opposite the given angle,
- \( a \) and \( b \) are the other two sides of the triangle,
- \( C \) is the angle opposite the side \( x \).

In this problem:
- \( a = 12 \) cm,
- \( b = 15 \) cm,
- \( C = 85^\circ \).

Let's plug in the values:

\[
x^2 = 12^2 + 15^2 - 2 \times 12 \times 15 \times \cos(85^\circ)
\]

Now let's calculate it step by step.

\[
x^2 = 144 + 225 - 360 \times \cos(85^\circ)
\]
\[
x^2 = 369 - 360 \times \cos(85^\circ)
\]
\[
\cos(85^\circ) \approx 0.0872
\]
\[
x^2 = 369 - 360 \times 0.0872
\]
\[
x^2 = 369 - 31.392
\]
\[
x^2 = 337.608
\]
\[
x = \sqrt{337.608} \approx 18.4 \text{ cm}
\]

So, the value of \( x \) correct to one decimal place is **18.4 cm**.

Would you like more details on any step or have any questions? Here are five related questions you might find interesting:

1. How does the Law of Cosines relate to the Pythagorean theorem?
2. What are other applications of the Law of Cosines?
3. How can the Law of Sines be used in solving triangle problems?
4. What happens if the angle in the Law of Cosines is \(90^\circ\)?
5. How do you solve for an angle given all three sides using the Law of Cosines?

**Tip:** In trigonometry, knowing when to apply the Law of Sines vs. the Law of Cosines can make solving problems much easier!

Find the value of x in a triangle with sides 12 cm, 15 cm, and an included angle of 85°, using the Law of Cosines. Give your answer correct to 1 decimal place.

Learn how to find the value of side x in a triangle with sides of 12 cm and 15 cm and an included angle of 85°. This trigonometry problem uses the Law of Cosines and is ideal for students in Grades 9-11.

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