Geometry and Trigonometry
Geometry and Trigonometry combines synthetic geometry proofs, constructions, trigonometric rules and identities, radian measure, and three-dimensional problems. Higher Level candidates must be able to prove specified theorems and apply them in unfamiliar diagrams.
Core rules
- Circle theorems include: the angle at the centre is twice the angle at the circumference; angles in the same segment are equal; opposite angles of a cyclic quadrilateral sum to (180^\circ); a tangent is perpendicular to the radius at the point of contact; the alternate segment theorem; a perpendicular from the centre to a chord bisects the chord.
- Constructions must show arcs and construction lines; do not erase them.
- Sine rule: (\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}). Cosine rule: (a^2=b^2+c^2-2bc\cos A). Area of a triangle: (\frac12 ab\sin C).
- Radians: (\pi) radians (=180^\circ). Arc length (s=r\theta); sector area (=\frac12 r^2\theta), with (\theta) in radians.
- Identities: (\sin^2\theta+\cos^2\theta=1), (\tan\theta=\frac{\sin\theta}{\cos\theta}). When solving trigonometric equations, find all solutions in the stated interval.
Worked example
In a triangle, (b=8), (c=7) and (A=60^\circ). Find (a). By the cosine rule, (a^2=8^2+7^2-2(8)(7)\cos60^\circ=64+49-56=57). Therefore (a=\sqrt{57}\approx7.55).
Traps
Using the sine rule in the ambiguous SSA case without considering a second triangle; leaving the calculator in radian mode for degree questions; forgetting the second solution for equations such as (\sin\theta=k); using degrees in (s=r\theta); omitting the given/prove structure in a theorem proof.