Revise Trigonometry: sine and cosine rules for IB Mathematics: Analysis and Approaches (3.2)

Before reaching for the new rules, always check whether the triangle already has a right angle — if it does, the faster tools still work. In a right-angled triangle, label the sides relative to the angle $\theta$ you care about: the hypotenuse (opposite the right angle), the opposite (across from $\theta$) and the adjacent (next to $\theta$). Then SOH-CAH-TOA applies, and Pythagoras relates the three sides.

$\sin\theta=\dfrac{\text{opp}}{\text{hyp}}, \quad \cos\theta=\dfrac{\text{adj}}{\text{hyp}}, \quad \tan\theta=\dfrac{\text{opp}}{\text{adj}}$ $\text{and} \quad \text{hyp}^2=\text{opp}^2+\text{adj}^2 \quad (\text{Pythagoras}).$

The sine rule links each side to the sine of its opposite angle: across a whole triangle the ratio of a side to the sine of the angle facing it is constant. So a single complete pair — one side with its opposite angle — is the 'key' that unlocks the rest, provided you have one further piece of information.

For a triangle with angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$: finding a side: $\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}$ finding an angle: $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$

The sine rule carries one famous trap. When you are given two sides and a non-included angle (SSA) and use the sine rule to find an angle, there may be TWO triangles that fit the data. This happens because $\sin\theta=\sin(180^\circ-\theta)$: your calculator's $\arcsin$ hands you the acute angle, but the obtuse partner can be equally valid. Always test it.

The cosine rule is Pythagoras' theorem with a correction term for triangles that are not right-angled — indeed if $A=90^\circ$ then $\cos A=0$ and it collapses back to $a^2=b^2+c^2$. Use it when there is no starting pair: a side sandwiched between two known sides (SAS), or all three sides known (SSS).

For a triangle with angles $A$, $B$, $C$ and opposite sides $a$, $b$, $c$: finding a side: $a^2=b^2+c^2-2bc\cos A$ finding an angle (rearranged): $\cos A=\dfrac{b^2+c^2-a^2}{2bc}$

When you know two sides and the angle between them, you do not need the base-and-height picture at all: the area is half the product of the two sides and the sine of their included angle. The word 'included' is doing all the work — the angle must sit between the two sides you use, or the formula is wrong.

Area $=\dfrac{1}{2}ab\sin C$, where $C$ is the angle INCLUDED between sides $a$ and $b$.

These rules come alive in navigation and surveying. A bearing is an angle measured clockwise from north, always written with three figures (for example $070^\circ$ or $250^\circ$). By drawing a north line at each point and using alternate angles between the parallel north lines, bearings convert into the interior angles of a triangle. Similarly, the angle of elevation is measured up from the horizontal to an object, and the angle of depression down from the horizontal; the two are equal alternate angles, which lets you carry an angle from an observer to a distant point.

On Paper 1 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 that a right method carried out on a wrong earlier number still earns the marks that depend on it, and 'ISW' (ignore subsequent working) means a correct answer is not un-awarded if a student then writes something clumsy. 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.