Let's solve the two parts of the problem shown in the image:

Part (a): Write the value of $\sin \theta$ as a fraction

In a right-angled triangle, $\sin \theta$ is defined as:

$\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

From the diagram, we see:

• The opposite side to angle $\theta$ is 13 cm.
• The hypotenuse is 18 cm.

Thus, the value of $\sin \theta$ is:

$\sin \theta = \frac{13}{18}$

Part (b): Calculate the size of angle $\theta$

To find the size of the angle $\theta$, we use the inverse sine function ($\sin^{-1}$):

$\theta = \sin^{-1} \left( \frac{13}{18} \right)$

Let's calculate the value of $\theta$ to 1 decimal place.The size of angle $\theta$ is approximately $46.2^\circ$, rounded to one decimal place.

Summary:

• (a) $\sin \theta = \frac{13}{18}$
• (b) $\theta \approx 46.2^\circ$

Would you like more details on any of the steps? Here are five related questions for further exploration:

1. How would you find $\cos \theta$ and $\tan \theta$ in this triangle?
2. How does the Pythagorean theorem relate to this triangle?
3. What is the sine rule, and how could it apply to non-right triangles?
4. How do we use trigonometric ratios in 3D shapes?
5. What are some real-life applications of using sine in right-angled triangles?

Tip: Always ensure your calculator is set to the correct mode (degrees or radians) when calculating trigonometric values.