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.