How Do You Use Bearings in a Trigonometry Problem?

Use bearings by drawing north lines, measuring each direction clockwise from north, and converting the bearings into interior angles of a triangle. Then apply right-angled trigonometry, the sine rule, or the cosine rule as appropriate.

A bearing is measured clockwise from north and written using three figures, such as $060^\circ$. This content belongs to IB Mathematics: Applications and Interpretation SL 3.3: Applications of trigonometry.

Choose the method from the information available: use right-angled trigonometry when the triangle contains a right angle; the sine rule when you know an opposite side-angle pair; and the cosine rule for two sides and the included angle, or three sides. If north lines create supplementary angles, check whether the triangle's angle is $180^\circ-\theta$.

For example, a boat travels $10\text{ km}$ from $A$ to $B$ on a bearing of $060^\circ$. It then travels $8\text{ km}$ from to on a bearing of .

The bearing of $A$ from $B$ is the reverse of $060^\circ$:

$$060^\circ+180^\circ=240^\circ.$$

Therefore, the angle inside triangle $ABC$ at $B$ is

$$240^\circ-140^\circ=100^\circ.$$

Using the cosine rule,

$$AC^2=10^2+8^2-2(10)(8)\cos(100^\circ),$$

so

$$AC=13.8\text{ km}$$

to three significant figures.

A common misconception is to use the difference between the two original bearings directly. Bearings are measured from separate north lines, so first determine the relevant back bearing or use parallel-line angle relationships.

In an IB response, include a labelled diagram, show how the interior angle was obtained, and write the trigonometric expression before the GDC value. Keep the GDC in degree mode at SL and write final bearings using three figures. This applies to SL Paper 1 and Paper 2, both of which require a GDC.