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The triangle shown in the image has two known angles (60° and 47°) and a known side of 3 cm. We are asked to find the side length \( z \), which is opposite the 60° angle.

### Steps to Solve:
We will apply the **Law of Sines**, which is expressed as:

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

Where:
- \( a, b, c \) are the side lengths opposite the angles \( A, B, C \) respectively.

In this case:
- \( z \) is opposite the 60° angle.
- The 3 cm side is opposite the 47° angle.

First, we need to calculate the third angle, which can be found using the angle sum property of a triangle:

\[
\text{Third angle} = 180^\circ - 60^\circ - 47^\circ = 73^\circ
\]

Now, using the Law of Sines, we set up the equation:

\[
\frac{z}{\sin(60^\circ)} = \frac{3}{\sin(47^\circ)}
\]

Solving for \( z \):

\[
z = \frac{3 \times \sin(60^\circ)}{\sin(47^\circ)}
\]

I'll calculate the value of \( z \).The value of \( z \) is approximately **3.6 cm**, correct to one decimal place.

Would you like further details or have any additional questions?

Here are some related questions:
1. How do you apply the Law of Cosines in solving triangles?
2. What are some other applications of the Law of Sines?
3. How can you determine if a triangle is obtuse, acute, or right-angled?
4. How do you solve for an unknown angle using the Law of Sines?
5. What are the limitations of the Law of Sines in solving triangles?

**Tip:** Always check if a triangle is solvable with the given information before using trigonometric laws.

Find the value of z in the triangle given angles and side lengths.

Discover how to find the side length of a triangle using the Law of Sines with a known side and two angles. This problem includes step-by-step instructions on applying the Law of Sines and the Angle Sum Property to solve for unknown side lengths in trigonometry.

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