3.2 Trigonometry in trianglesIB Maths: Applications and Interpretation HL: Revision notes

In a right-angled triangle, relative to an angle $\theta$, the hypotenuse is opposite the right angle, the opposite side faces $\theta$ and the adjacent side is next to $\theta$: $\sin\theta=\frac{\text{opp}}{\text{hyp}},\qquad\cos\theta=\frac{\text{adj}}{\text{hyp}},\qquad\tan\theta=\frac{\text{opp}}{\text{adj}}.$ Use Pythagoras' theorem $a^2+b^2=c^2$ when two sides are known and a third is needed. To find an angle, use the inverse functions $\sin^{-1}$, $\cos^{-1}$, $\tan^{-1}$ (link to SL 2.2). Check that your GDC is in degree mode. Example: a 5 m ladder makes $70^\circ$ with the ground, so it reaches $5\sin70^\circ=4.70$ m up a wall. A 3 m rise on a 7 m slope gives $\theta=\sin^{-1}\left(\frac37\right)=25.4^\circ$.

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 the angle opposite it (a complete pair) plus one more side or angle. To find an angle, use the form $\frac{\sin A}{a}=\frac{\sin B}{b}$. Example: $a=8$, $A=40^\circ$, $B=75^\circ$. Then $b=\frac{8\sin75^\circ}{\sin40^\circ}=12.0$. Find the third angle first using the angle sum of $180^\circ$. The ambiguous case of the sine rule is not required at SL.

Use the cosine rule when you know two sides and the included angle (the angle between them), or all three sides: $c^2=a^2+b^2-2ab\cos C,\qquad\cos C=\frac{a^2+b^2-c^2}{2ab}.$ The side $c$ on the left is opposite the angle $C$. If $\cos C<0$, the angle is obtuse. Example: $a=7$, $b=9$, $C=65^\circ$ gives $c^2=49+81-126\cos65^\circ=76.75$, so $c=8.76$. For sides 5, 6, 7 the largest angle is opposite 7: $\cos C=\frac{25+36-49}{60}=0.2$, so $C=78.5^\circ$.

The area of a triangle is $A=\tfrac12ab\sin C,$ where $C$ is the angle between sides $a$ and $b$. It works for any triangle, including obtuse ones. Example: sides 8 and 11 with included angle $35^\circ$ give $A=\frac12(8)(11)\sin35^\circ=25.2$. You can combine rules: use the cosine rule to find an angle, then the area formula, or split a quadrilateral into two triangles and add their areas.

Sketch a well-labelled diagram first, then choose:

• right angle in the triangle: SOH CAH TOA or Pythagoras
• a side and its opposite angle, plus one more piece of data: sine rule
• two sides and the included angle, or three sides: cosine rule
• two sides and the included angle, and you need area: $\frac12ab\sin C$. For problems with several triangles, such as a quadrilateral field, find a shared side first, then use it in the second triangle. Keep full calculator values and round only at the end, to 3 significant figures.