3.2 Right-angled and non-right-angled trigonometryIB Maths: Analysis and Approaches SL: Revision notes

In a right-angled triangle, label the sides relative to the angle $\theta$ you are using: the hypotenuse (opposite the right angle), the opposite side and the adjacent side. $\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad\cos\theta=\frac{\text{adj}}{\text{hyp}},\quad\tan\theta=\frac{\text{opp}}{\text{adj}}.$ Use Pythagoras for a missing side when two sides are known, and the inverse functions $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ for an angle. In a 5–12–13 triangle with the right angle at $B$, $\tan A=\frac{12}{5}$ and $\sin C=\frac{5}{13}$. Note that $\sin A=\cos C$, because the side opposite $A$ is adjacent to $C$.

For any triangle with sides $a,b,c$ opposite angles $A,B,C$: $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.$ Use the sine rule when you know a side and its opposite angle, plus one more side or angle. Finding an angle is easier with the reciprocal form $\frac{\sin A}{a}=\frac{\sin B}{b}$. Example: $XY=10$, $X=45^\circ$, $Y=75^\circ$. First $Z=60^\circ$, then $YZ=\frac{10\sin45^\circ}{\sin60^\circ}=\frac{10\sqrt6}{3}$. (The ambiguous case, where two triangles fit the data, is covered in 3.5.)

Use the cosine rule when you know two sides and the included angle (to find the third side), or all three sides (to find an angle): $c^2=a^2+b^2-2ab\cos C,\qquad \cos C=\frac{a^2+b^2-c^2}{2ab}.$ The side on the left, $c$, must be opposite the angle $C$. If $\cos C<0$ the angle is obtuse; the cosine rule handles this automatically, which the sine rule cannot. Example: $PQ=8$, $PR=5$, $P=60^\circ$ gives $QR^2=64+25-40=49$, $QR=7$.

$\text{Area}=\frac12ab\sin C,$ where $C$ is the angle between sides $a$ and $b$. For $PQ=8$, $PR=5$ and $P=60^\circ$: area $=\frac12(8)(5)\frac{\sqrt3}{2}=10\sqrt3$. Two useful links:

• Equating $\frac12ab\sin C$ with $\frac12\times$ base $\times$ height gives the perpendicular distance from a vertex to the opposite side.
• A line from a vertex to the midpoint of the opposite side (a median) splits a triangle into two equal areas, because both parts share the same height.

• Right angle present: SOH CAH TOA and Pythagoras.
• A side and its opposite angle known: sine rule.
• Two sides and the included angle, or three sides: cosine rule.
• Area: $\frac12ab\sin C$.

Keep full calculator values between steps and round only at the end (3 s.f. for lengths, 1 d.p. for angles in degrees). In a non-calculator question, use exact values such as $\sin60^\circ=\frac{\sqrt3}{2}$ and $\cos60^\circ=\frac12$.