To calculate angle $\theta$ using the sine rule, follow these steps:

The sine rule states: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$

Here:

• $A = 72^\circ$
• $a = 10 \, \text{cm}$
• $b = 7 \, \text{cm}$
• $\theta = B$

From the sine rule: $\frac{\sin 72^\circ}{10} = \frac{\sin \theta}{7}$

Step 1: Solve for $\sin \theta$

$\sin \theta = \frac{\sin 72^\circ \cdot 7}{10}$

First, calculate $\sin 72^\circ$ (using a calculator): $\sin 72^\circ \approx 0.9511$

Now: $\sin \theta = \frac{0.9511 \cdot 7}{10} = \frac{6.6577}{10} \approx 0.6658$

Step 2: Find $\theta$ using the inverse sine

$\theta = \sin^{-1}(0.6658)$

Using a calculator: $\theta \approx 41.9^\circ$

Final Answer:

$\theta \approx 41.9^\circ \, (\text{to 1 decimal place})$

Would you like me to go into more detail or clarify any part of this solution?

Related Questions:

1. How can the sine rule be applied to non-right-angled triangles?
2. What is the cosine rule, and how does it differ from the sine rule?
3. How can you verify your answer geometrically?
4. What is the relationship between all angles in a triangle, and how can this check your work?
5. Can the sine rule determine sides instead of angles?

Tip:

Always double-check that your calculated sine values stay within the range $[0, 1]$, as any number outside this range is invalid for sine calculations!