https://file.aiquickdraw.com/mathbot/file/1727218407568ji6x3j0m.jpg

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.

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

Learn how to find the sine of angle θ and calculate its size in degrees using the given right-angled triangle. With an opposite side of 13 cm and a hypotenuse of 18 cm, we calculate sin(θ) = 13/18 and use the inverse sine function to find θ ≈ 46.2°.

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