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Use the law of sines to find the indicated side x or reference angle (how to get answer: 44°)

We are given a triangle with the following details:

- One angle, \( \angle A = 120^\circ \), is provided.
- Two sides: \( AB = 36 \) and \( AC = 45 \).
- We are tasked to find angle \( \theta \) (the angle at \( C \)) using the Law of Sines.

The Law of Sines states:

\[
\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}
\]

Where:
- \( A \) is the angle opposite side \( a \),
- \( B \) is the angle opposite side \( b \),
- \( C \) is the angle opposite side \( c \).

From the diagram:
- Angle \( A = 120^\circ \),
- Side \( AB = 36 \),
- Side \( AC = 45 \),
- \( \theta \) is the unknown angle at \( C \).

Let’s proceed step by step.

### Step 1: Using the Law of Sines to find angle \( C \)

We can apply the Law of Sines as follows:

\[
\frac{AC}{\sin A} = \frac{AB}{\sin C}
\]

Substitute the known values into the equation:

\[
\frac{45}{\sin 120^\circ} = \frac{36}{\sin \theta}
\]

Now let's solve for \( \theta \). I will compute this next.The angle \( \theta \) at point \( C \) is approximately \( 44^\circ \), as expected.

Would you like more details on the steps, or do you have any additional questions? 

Here are 5 related questions to expand your understanding:
1. How can the Law of Cosines be used in a triangle like this one?
2. What are some real-life applications of the Law of Sines?
3. How can you verify this result using another method, such as the Law of Cosines?
4. What would change if one of the given sides were longer?
5. How can you determine the remaining angle \( B \) in this triangle?

**Tip**: Always remember that the Law of Sines applies to any type of triangle, but ensure that the sine values are within the valid range of the arcsine function.

Use the law of sines to find the indicated side x or reference angle (how to get answer: 44°).

Learn how to use the Law of Sines to calculate an unknown angle in a triangle given a 120° angle and two sides measuring 36 and 45. Step-by-step solution with formula application. Suitable for students in Grades 10-12.

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