The diagram shows three points, A, B and L on a horizontal field. $AL = 37.5\ \mathrm{m}, AB = 60\ \mathrm{m}$ and $\angle LAB = 63^\circ$.

1. Identify and apply the cosine rule for the unknown side $LB$.

In $\triangle ALB$, $AL=37.5$, $AB=60$ and $\angle LAB=63^\circ$.

By the cosine rule,
$LB^2=AL^2+AB^2-2(AL)(AB)\cos \angle LAB.$

2. Substitute the given values correctly.

$LB^2=37.5^2+60^2-2(37.5)(60)\cos 63^\circ$

3. Evaluate and round the length appropriately.

$LB^2=2963.28\ldots$
$LB=\sqrt{2963.28\ldots}=54.436\ldots$

So, to 3 significant figures,
$LB=54.4\text{ m}.$

1. Set up the sine rule with the correct opposite side-angle pairs.

In $\triangle ALB$, $LB$ is opposite $\angle LAB=63^\circ$, and $AL$ is opposite $\angle LBA$.

$\frac{\sin \angle LBA}{37.5}=\frac{\sin 63^\circ}{56.2}$

2. Calculate the angle and round appropriately.

$\sin \angle LBA=\frac{37.5\sin 63^\circ}{56.2}$

$\angle LBA=\sin^{-1}\left(\frac{37.5\sin 63^\circ}{56.2}\right)$

$\angle LBA=36.5^\circ\text{ approximately}$

1. Identify the right triangle and choose the tangent ratio.

Let the height of the tree be $LT$.

In right-angled triangle $ALT$,
$\tan 10^\circ=\frac{LT}{AL}$
$\tan 10^\circ=\frac{LT}{37.5}$

2. Calculate the height of the tree.

$LT=37.5\tan 10^\circ$
$LT=6.611\ldots$

So the height of the tree is $6.61$ m, to 3 significant figures.

1. Identify the right triangle and choose the appropriate trigonometric ratio.

Let $d$ be the distance the bird must fly.

The angle of depression is $60^\circ$, so the angle between the bird's horizontal line and the line of sight to the prey is $60^\circ$.

The vertical height is $3$ m, and this is opposite the $60^\circ$ angle. The flying distance $d$ is the hypotenuse.

$\sin 60^\circ = \frac{3}{d}$

2. Calculate the slant distance and round suitably.

$d = \frac{3}{\sin 60^\circ}$

$d = 3.464\ldots$

$d = 3.46\text{ m}$