The diagram below shows a spotlight at point B, which is mounted 12 m directly above point C, at the front of the field. The spotlight swings 44,3 degrees from the vertical to illuminate another point, D, the midpoint of AC.

In $\triangle BCD$, we have a right angle at C. We are given $BC = 12$ m and $\angle CBD = 44.3^\circ$. We need to find the length of $DC$.

The tangent of an angle in a right-angled triangle is the ratio of the opposite side to the adjacent side.

The length of DC is $\boxed{11.71 \text{ m}}$.

3.1.2 Calculate through how many degrees the spotlight must swing in order to illuminate point A (i.e. calculate $\theta$).

Since D is the midpoint of AC, $AC = 2 \times DC$.

In $\triangle ABC$, we have a right angle at C. We know $BC = 12$ m and $AC = 23.4168$ m.

The angle $\theta$ is the difference between $\angle CBA$ and $\angle CBD$.

The spotlight must swing through $\boxed{18.57^\circ}$.

Question 3.2

Assume that ABCD is a rectangle in a horizontal plane and EB is a vertical line segment. The angle of elevation from D to E is $\angle EDA = 22^\circ$.

Calculate the length of BE.

In right-angled $\triangle EAD$ (right-angled at A), we are given $DE = 800$ cm and $\angle EDA = 22^\circ$.
We can find $AE$ using the sine function:

Since ABCD is a rectangle, $AB = DC = 155$ cm.
The length $BE$ is the sum of $AB$ and $AE$.

The length of BE is $\boxed{454.68 \text{ cm}}$.

Determine the distance of BC.

In right-angled $\triangle EAD$, we can find $AD$ using the cosine function:

Since ABCD is a rectangle, $BC = AD$.

The distance of BC is $\boxed{741.75 \text{ cm}}$.