Let's break down the problem from the image:

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

For a right-angled triangle: $\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}$ From the diagram:

• Opposite side to $\theta$ = 13 cm
• Hypotenuse = 18 cm

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

Part (b): Calculate the size of angle $\theta$ (in degrees, 1 decimal place)

We need to find the angle $\theta$ using: $\sin \theta = \frac{13}{18}$

To find $\theta$, take the inverse sine (arcsin) of the fraction: $\theta = \sin^{-1}\left(\frac{13}{18}\right)$

Using a calculator: $\theta \approx \sin^{-1}(0.7222) \approx 46.0^\circ$

Final Answers:

• (a) $\sin \theta = \frac{13}{18}$
• (b) $\theta \approx 46.0^\circ$ (to 1 decimal place)

Would you like me to explain any of the steps in more detail or check further calculations?

Here are 5 related questions to deepen your understanding:

1. What are the other trigonometric ratios for this triangle (cosine and tangent)?
2. How can we use the Pythagorean theorem to verify the hypotenuse length?
3. What is the meaning of the inverse sine function?
4. What happens if you use radians instead of degrees to calculate the angle?
5. Can you convert the sine value into a decimal and still maintain accuracy?

Tip: In trigonometry, it's good practice to always label the sides relative to the angle of interest (opposite, adjacent, hypotenuse).