To determine the angle $\theta$ in the triangle, we can apply the Sine Rule. Here's how we proceed:

Step 1: Identify the known values

• Angle $54^\circ$ is given.
• The two sides adjacent to this angle are $7 \, \text{cm}$ and $6 \, \text{cm}$.

We will use the Sine Rule, which states:

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

where:

• $a$, $b$, and $c$ are the lengths of the sides.
• $A$, $B$, and $C$ are the angles opposite those sides.

Step 2: Apply the Sine Rule

From the diagram, we have:

• Angle $A = 54^\circ$
• Side $a = 6 \, \text{cm}$
• Side $b = 7 \, \text{cm}$
• Angle $\theta = B$

Using the Sine Rule, we set up the following equation:

$\frac{7}{\sin 54^\circ} = \frac{6}{\sin \theta}$

Step 3: Solve for $\theta$

Rearranging the equation:

$\sin \theta = \frac{6 \sin 54^\circ}{7}$

Now we can calculate the value for $\theta$.

Let me compute that for you. math

Given values

side_a = 6 # cm side_b = 7 # cm angle_A = 54 # degrees

Convert angle_A to radians

angle_A_rad = math.radians(angle_A)

Using the Sine Rule to find sin(theta)

sin_theta = (side_a * math.sin(angle_A_rad)) / side_b

Calculate theta in radians and then convert to degrees

theta_rad = math.asin(sin_theta) theta_deg = math.degrees(theta_rad)

Round to 3 significant figures

theta_deg_rounded = round(theta_deg, 3)

theta_deg_roundedThe angle $\theta$ is approximately $43.9^\circ$, correct to three significant figures.

Would you like any further clarification on this solution? Here are some related questions to explore:

1. How do you apply the Sine Rule in non-right angled triangles?
2. What other methods can be used to solve for angles in triangles?
3. Can the Cosine Rule be applied in this scenario? If so, how?
4. What is the difference between the Sine Rule and the Cosine Rule?
5. How would the solution change if the side lengths were different?

Tip: When applying the Sine Rule, always ensure that the angle you're using corresponds to the side you're referring to!