A triangle has a 56-degree angle, formed by a 10-inch side and an x-inch side. Given

that the area of the triangle is 18 square inches, find x.

To find \( x \), we'll use the formula for the area of a triangle when two sides and the included angle are known:

\[
\text{Area} = \frac{1}{2}ab \sin(C)
\]

where:
- \( a = 10 \) inches,
- \( b = x \) inches (the unknown side),
- \( C = 56^\circ \),
- \(\text{Area} = 18\) square inches.

### Step-by-Step Solution

1. Substitute the known values into the area formula:

   \[
   18 = \frac{1}{2} \cdot 10 \cdot x \cdot \sin(56^\circ)
   \]

2. Simplify the equation:

   \[
   18 = 5x \cdot \sin(56^\circ)
   \]

3. Calculate \( \sin(56^\circ) \). Using a calculator or trigonometric table, we find:

   \[
   \sin(56^\circ) \approx 0.829
   \]

   Substitute this value back into the equation:

   \[
   18 = 5x \cdot 0.829
   \]

4. Simplify further:

   \[
   18 = 4.145x
   \]

5. Solve for \( x \) by dividing both sides by 4.145:

   \[
   x = \frac{18}{4.145} \approx 4.34
   \]

### Answer
The length of side \( x \) is approximately **4.34 inches**.

Would you like more details or have any questions about the process?

---

Here are 5 related questions to expand on this topic:

1. How do you find the area of a triangle if all three sides are known?
2. What is the Law of Sines, and how can it be used with triangles?
3. How can you find an unknown angle in a triangle if two sides and one angle are known?
4. How would the solution change if the angle given was obtuse instead of acute?
5. What are other applications of trigonometric functions in geometry?

**Tip:** When solving for unknowns in trigonometry, it's useful to keep the trigonometric values (like \(\sin(56^\circ)\)) precise until the final step to maintain accuracy.

A triangle has a 56-degree angle, formed by a 10-inch side and an x-inch side. Given that the area of the triangle is 18 square inches, find x.

Explore how to find the length of side x in a triangle with a 56-degree angle and an area of 18 square inches using trigonometric principles. This problem includes step-by-step calculations and relevant mathematical concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation