The diagonal and the longer side of a rectangle make an angle of 43.2. If the longer side is 12.6 cm, find the length of the shorter side.

Let $L$ be the longer side (adjacent to the angle) and $W$ be the shorter side (opposite the angle).
Given: $L = 12.6$ cm, $\theta = 43.2^\circ$.
We use the tangent function:

The length of the shorter side is approximately $\boxed{\text{11.83 cm}}$.

6. A rhombus has diagonals of length 12 cm and 7 cm respectively. Find the larger angle of the rhombus.

The diagonals of a rhombus bisect each other at right angles. This forms four congruent right-angled triangles. The legs of these triangles are half the lengths of the diagonals.
Half of diagonal 1: $x = \frac{12 \text{ cm}}{2} = 6$ cm.
Half of diagonal 2: $y = \frac{7 \text{ cm}}{2} = 3.5$ cm.

Let $\alpha$ be half of the larger angle of the rhombus. In one of the right-angled triangles, $\alpha$ is the angle opposite the longer half-diagonal ($x$) and adjacent to the shorter half-diagonal ($y$).

The larger angle of the rhombus is $2\alpha$.

The larger angle of the rhombus is approximately \boxed{\text{119.48^\circ}}.

7. Find the angle of elevation to the top of a 62 m high building from point A, which is at ground level 155m from its base. What is the angle of depression from the top of the building to A?

Let $h$ be the height of the building (62 m) and $d$ be the distance from point A to the base (155 m). Let $\theta_E$ be the angle of elevation.

The angle of elevation is approximately \boxed{\text{21.80^\circ}}.

The angle of depression from the top of the building to point A is equal to the angle of elevation from point A to the top of the building (alternate interior angles).
Therefore, the angle of depression is also approximately \boxed{\text{21.80^\circ}}.

8. The angle of depression from the top of a 180 m high vertical cliff to a boat B is 26. Find how far the boat is from the base of the cliff.

Let $h$ be the height of the cliff (180 m).
Let $d$ be the distance from the boat to the base of the cliff (horizontal distance).
The angle of depression from the top of the cliff to the boat is $26^\circ$. This means the angle of elevation from the boat to the top of the cliff is also $26^\circ$.
We use the tangent function:

The boat is approximately $\boxed{\text{369.09 m}}$ from the base of the cliff.