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This is a triangle problem where you are asked to determine the unknown side \( x \) and the unknown angle \( \theta \) to the nearest tenth of a unit. The triangle includes the following information:

- One angle is \( 50^\circ \).
- The sides adjacent to this angle are 4.0 cm and 4.5 cm.
- The unknown side is \( x \).
- The unknown angle is \( \theta \).

To solve this, we can use the Law of Cosines and Law of Sines. Let's start by finding side \( x \) using the Law of Cosines.

### Step 1: Use the Law of Cosines to find \( x \)
The Law of Cosines is given by:
\[
x^2 = a^2 + b^2 - 2ab \cdot \cos(C)
\]
Where:
- \( a = 4.0 \, \text{cm} \)
- \( b = 4.5 \, \text{cm} \)
- \( C = 50^\circ \)

Substituting in the values:
\[
x^2 = 4.0^2 + 4.5^2 - 2 \cdot 4.0 \cdot 4.5 \cdot \cos(50^\circ)
\]
\[
x^2 = 16 + 20.25 - 36 \cdot \cos(50^\circ)
\]
Now, compute \( \cos(50^\circ) \approx 0.6428 \):
\[
x^2 = 16 + 20.25 - 36 \cdot 0.6428
\]
\[
x^2 = 36.25 - 23.14
\]
\[
x^2 = 13.11
\]
\[
x \approx \sqrt{13.11} \approx 3.6 \, \text{cm}
\]

### Step 2: Use the Law of Sines to find \( \theta \)
Now that we know \( x \approx 3.6 \, \text{cm} \), we can apply the Law of Sines to find \( \theta \). The Law of Sines is:
\[
\frac{\sin(A)}{a} = \frac{\sin(B)}{b} = \frac{\sin(C)}{c}
\]
We already know \( C = 50^\circ \), \( a = 4.0 \, \text{cm} \), \( b = 4.5 \, \text{cm} \), and \( x = 3.6 \, \text{cm} \). So, using the Law of Sines:
\[
\frac{\sin(50^\circ)}{3.6} = \frac{\sin(\theta)}{4.5}
\]
\[
\sin(\theta) = \frac{4.5 \cdot \sin(50^\circ)}{3.6}
\]
Using \( \sin(50^\circ) \approx 0.766 \):
\[
\sin(\theta) = \frac{4.5 \cdot 0.766}{3.6} \approx \frac{3.447}{3.6} \approx 0.957
\]
\[
\theta = \sin^{-1}(0.957) \approx 73.8^\circ
\]

### Final Answers:
- \( x \approx 3.6 \, \text{cm} \)
- \( \theta \approx 73.8^\circ \)

Would you like more details on any of the steps? Let me know if you have any questions.

---

Here are five follow-up questions:
1. How would the calculation change if the angle \( C \) were larger than \( 50^\circ \)?
2. Can you use the Law of Cosines to find angle \( \theta \) directly instead of the Law of Sines?
3. How would you check the solution's consistency using the triangle's angle sum?
4. How can trigonometric values (e.g., \( \sin(50^\circ) \)) be approximated more precisely?
5. What are the other applications of the Law of Cosines in real-world problems?

**Tip:** The Law of Sines is particularly useful for solving non-right triangles when you know one angle and its opposite side!

Determine the values indicated with variables to the nearest tenth of a unit for a triangle with sides 4.0 cm, 4.5 cm, an angle of 50°, and unknowns x and θ.

Solve a triangle with sides 4.0 cm, 4.5 cm, an angle of 50°, and unknowns x and θ using the Law of Cosines and the Law of Sines. Step-by-step solutions provided. Suitable for high school students (Grades 10-12).

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