A rectangle has an area of \(91 \mathrm{~cm}^2\) and a length of \(13 \mathrm{~cm}\). Find the angle formed between the length and the diagonal. (to the nearest degree)

A cone has a diameter of \(20 \mathrm{~mm}\) and a slant height of \(26 \mathrm{~mm}\). Calculate the size of the vertical angle \(\theta\).

If \(\sin 30^{\circ}=\dfrac{1}{2}\), find as a surd the value of \(\tan 60^{\circ}\).

If \(\tan 45^{\circ}=1\), find as a surd the value of \(\cos 45^{\circ}\).

Find the distance between (to the nearest km)

a) checkpoints 1 and 2
b) checkpoint 2 and the finish. 

Triangle \(ABC\) is right - angled at \(B\), \(AC = 13.9 ~\text{cm}\) and angle \(C\) is \(66\) degrees. Find the length of \(BC\), correct to one decimal place.

A tree casts a shadow 20 m long. If the Sun's rays meet the ground at 25 degrees, find the height of the tree, correct to the nearest cm.

A \(7 ~\text{m}\) ladder is placed against a pole. If the ladder makes an angle of \({\rm{17}}^\circ \) with the pole, what is the distance between the base of the pole and bottom of the ladder (correct to 2 decimal places)?

A boat is anchored by a rope \(5.5~\text{m}\) long. If the anchor rope makes an angle of \(23\) degrees with the vertical, calculate the depth of the water (correct to one decimal place).