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.