https://file.aiquickdraw.com/mathbot/file/17255343719450yg7b2m2.png

Let's break down and solve both problems in the image.

### Question 3: Calculate the value of \( \theta \)
We are given a right-angled triangle \( ABC \), where:
- \( AB = 10 \) cm (hypotenuse),
- \( BC = 3 \) cm (one side),
- \( CD = 5 \) cm (another side).

We are asked to calculate \( \theta \), which is the angle at \( A \).

Using trigonometry, we can apply the cosine rule for right-angled triangles:
\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AB}
\]

Substitute the values:
\[
\cos(\theta) = \frac{3}{10} = 0.3
\]
Now, take the inverse cosine to find \( \theta \):
\[
\theta = \cos^{-1}(0.3)
\]
I'll calculate this for you.

### Question 4: Calculate the length of side \( x \)
This problem involves another triangle, where:
- One angle is \( 42^\circ \),
- Another angle is \( 51^\circ \),
- The side opposite \( 42^\circ \) is \( 8 \) cm,
- We are to calculate the length of side \( x \).

First, let's confirm that the third angle is \( 180^\circ - 42^\circ - 51^\circ = 87^\circ \).

We can use the **sine rule** to solve for \( x \):
\[
\frac{x}{\sin(51^\circ)} = \frac{8}{\sin(87^\circ)}
\]

Now we can rearrange to solve for \( x \):
\[
x = \frac{8 \times \sin(51^\circ)}{\sin(87^\circ)}
\]
Let me calculate both values for you.Here are the results:

### Question 3: 
The value of \( \theta \) is approximately **72.54°**.

### Question 4: 
The length of the side labeled \( x \) is approximately **6.23 cm**.

Would you like further explanation or have any questions?

Here are some related questions you could explore:
1. How do we calculate the remaining sides of a right triangle using trigonometric ratios?
2. How does the sine rule work in non-right-angled triangles?
3. Can we use the cosine rule to solve for angles in non-right triangles?
4. What are other methods to find angles besides using trigonometric functions?
5. How can we use the tangent function to solve right triangle problems?

### Tip:
When using trigonometric functions, always ensure your calculator is in the correct mode (degrees or radians) for accurate results.

Calculate the value of θ in the right-angled triangle ABC, where AB = 10 cm (hypotenuse), BC = 3 cm (one side), and CD = 5 cm (another side). Calculate the length of side x in another triangle, where one angle is 42°, another angle is 51°, and the side opposite 42° is 8 cm.

Learn how to calculate the angle θ in a right-angled triangle using cosine and the length of side x in another triangle using the sine rule. This problem explores trigonometric ratios and the sine rule, suitable for students in Grades 8-10.

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