Geometry and Trigonometry
What the domain tests
Geometry and trigonometry in AI HL combines triangle trigonometry, circles and sectors, 3D geometry, and vectors. The exam tests selection of the correct rule, clear diagram labelling, and vector reasoning in two or three dimensions. Radian measure is expected in arc and sector formulas.
Core rules
- Sine rule: a/sin A = b/sin B = c/sin C.
- Cosine rule: a²=b²+c²-2bc cos A.
- Area of a triangle: 1/2 bc sin A.
- Arc length s=rθ; sector area 1/2 r²θ, with θ in radians.
- Vector magnitude |a|=sqrt(a_1²+a_2²+a_3²).
- Scalar product a·b=|a||b|cos θ; perpendicular vectors have a·b=0.
- Line: r=a+λb, where a is a point and b a direction vector.
Worked example
Find the angle between a=(2,1,-2) and b=(1,0,3). a·b=2(1)+1(0)+(-2)(3)=2-6=-4. |a|=sqrt(4+1+4)=3 and |b|=sqrt(1+0+9)=sqrt10. Thus cos θ=-4/(3sqrt10)≈-0.4216, so θ≈114.9°. The negative scalar product immediately signals an obtuse angle; the calculation confirms it.
Traps
Using the sine rule when the given information is SAS; missing the ambiguous sine-rule case; using degrees in s=rθ; treating a direction vector as a position vector; forgetting that a negative dot product gives an obtuse angle; losing a sign in 3D coordinates.