Revise Applications: bearings, elevation and gradients for IB Mathematics: Applications and Interpretation (3.3)

A bearing describes the direction of one point from another. The convention used throughout the exam is to measure the angle from the north line in a clockwise direction, and to write it with three figures. So due east is $090^\circ$, south is $180^\circ$, and a direction thirty degrees east of north is written $030^\circ$, not $30^\circ$.

The single most important step in a bearing problem is turning the given bearings into an interior angle of the triangle you can actually solve. Once you have that angle plus the two leg distances, the problem is an ordinary sine- or cosine-rule triangle.

These terms describe a line of sight relative to the horizontal. The angle of elevation is measured UP from the horizontal to an object above you; the angle of depression is measured DOWN from the horizontal to an object below you. A key fact: the angle of depression from an observer at $A$ down to an object at $B$ equals the angle of elevation from $B$ up to $A$, because the two horizontal lines are parallel and the sight line makes equal alternate (Z) angles with them. That equality lets you transfer an angle from one end of the sight line to the other.

The gradient measures the steepness of a slope. It is the ratio of the vertical change (the 'rise') to the horizontal change (the 'run'), and can be written as a fraction, a decimal or a percentage. Crucially, the run is the HORIZONTAL distance, not the slant distance travelled along the slope. The gradient is also linked to the angle of inclination $\theta$ — the angle the slope makes with the horizontal — by $m=\tan\theta$.

$m=\dfrac{\text{rise}}{\text{run}}=\tan\theta, \qquad \theta=\arctan(m)$ $\text{percentage gradient}=m\times 100\%$

On Paper 2 the marks are analytic: each is tied to a specific line of working — a method mark ($M$) for the correct approach, or an accuracy mark ($A$) for the right value, where an $A$ mark is DEPENDENT on the $M$ mark before it. Follow-through (FT) means a correct method applied to a wrong earlier number still earns the marks that depend on it, so one slip is punished once, not twice. But this protection only exists if the method is on the page. Study how each of the five marks below is earned by a specific line.