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The problem shows a right-angled triangle with one angle of \(52^\circ\), the hypotenuse of 13.8 meters, and we are tasked with finding the length of the side \(x\).

To solve for \(x\), we can use trigonometric functions. Since we have the hypotenuse and need to find the adjacent side (which is \(x\)), we use the cosine function:

\[
\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}
\]

Here, \(\theta = 52^\circ\) and the hypotenuse is 13.8 m. Therefore,

\[
\cos(52^\circ) = \frac{x}{13.8}
\]

Solving for \(x\):

\[
x = 13.8 \times \cos(52^\circ)
\]

Let me calculate that.The length of \(x\) is approximately 8.50 meters when rounded to two decimal places.

Would you like more details or have any further questions?

Here are 5 related questions to extend this concept:

1. How would you calculate the opposite side using trigonometry in this triangle?
2. What trigonometric function relates the opposite side and the hypotenuse in a right triangle?
3. How do you determine the other angle of this triangle besides \(52^\circ\)?
4. Can you explain how the sine and cosine of complementary angles are related?
5. What happens to the cosine value as the angle increases towards \(90^\circ\)?

**Tip:** In any right-angled triangle, cosine is always the ratio of the adjacent side to the hypotenuse. The larger the angle, the smaller the adjacent side, which is why cosine decreases as the angle increases.

Using trigonometry, calculate the length x in the right-angled triangle, given the hypotenuse is 13.8 meters and one angle is 52°. Give your answer in meters to 2 decimal places.

Learn how to use trigonometry to calculate the adjacent side x in a right-angled triangle with a 52° angle and a hypotenuse of 13.8 meters. This example demonstrates the use of the cosine function to solve for x, which is approximately 8.50 meters.

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