What are the answers to these trigonometry problems involving angles of elevation, bearings, and finding the length of a side in a right-angled triangle?

Bearings are measured clockwise from the North line, usually expressed as a three-digit number.

The diagram shows a North line at point A. The angle from the North line clockwise to the line segment AP is given as $56^\circ$.

Since $56^\circ$ is a two-digit number, we add a leading zero to make it three digits.
Answer:
The bearing of P from A is $\boxed{\mathbf{056^\circ}}$.

3. Find the exact length of AC.

Angle C is $90^\circ$.
Angle B is $30^\circ$.
Length of BC (adjacent side to angle B) is 12.
We need to find the length of AC (opposite side to angle B).

Since we have the opposite side and the adjacent side relative to angle B, we use the tangent function:

The exact value of $\tan(30^\circ)$ is $\frac{1}{\sqrt{3}}$ or $\frac{\sqrt{3}}{3}$.

Answer:
The exact length of AC is $\boxed{\mathbf{4\sqrt{3}}}$.

4. Q is due East of R and due South of P. The bearing of R from P is 230°. Show this information on the diagram.

To show this information on a diagram, you would draw the points and lines as follows:

Draw a North line from P.

Draw a line segment in this direction. Point R will be located somewhere along this line segment.

Point Q will be located on this line.

Point Q will be located on this line.

Ensure that Q is to the right of R and below P.

Diagram Description:

• Draw point P.
• Draw a vertical line upwards from P, labeled 'N' for North.
• From the North line at P, draw a line segment PR such that the angle measured clockwise from the North line to PR is $230^\circ$.
• Draw a vertical line downwards from P. Point Q lies on this line.
• Draw a horizontal line to the right from R. Point Q lies on this line.
• The point where the vertical line from P and the horizontal line from R intersect is point Q.
• This arrangement will show P, Q, and R forming a right-angled triangle with the right angle at Q, where PQ is a vertical line and RQ is a horizontal line.

Answer:
The information is shown by drawing points P, Q, and R such that:

• A line from P to R makes an angle of $230^\circ$ clockwise from the North line at P.
• Q is directly below P (on a vertical line).
• Q is directly to the right of R (on a horizontal line).