https://file.aiquickdraw.com/mathbot/file/1729181319558diev50ll.jpg

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).

a) Write the value of sin(θ) for the right-angled triangle below as a fraction. b) Using your answer to part a), work out the size of angle θ. Give your answer in degrees to 1 d.p.

Solve for angle θ in a right-angled triangle where the opposite side is 13 cm and the hypotenuse is 18 cm. Learn how to apply the sine function and inverse sine to find the angle in degrees to 1 decimal place.

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