Geometry and trigonometry
What Geometry and trigonometry tests
This domain tests spatial and trigonometric reasoning: triangle rules, the unit circle, trigonometric identities and equations, vectors, and basic 3D measurement. The practice expects exact values where possible and clear selection of a rule.
Core rules
- Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a^2 = b^2 + c^2 − 2bc cos A. Area: (1/2)ab sin C.
- Unit circle: sin^2 θ + cos^2 θ = 1, tan θ = sin θ/cos θ. Solve equations in the given interval using period and symmetry; use radians unless degrees are stated.
- Vectors: a·b = |a||b| cos θ = a_1b_1 + a_2b_2 + a_3b_3; perpendicular vectors have dot product zero; a line can be written r = a + λb.
- For 3D solids, use the formula booklet and identify the perpendicular height; slant height is not vertical height.
Worked example
In triangle ABC, AB = 7, AC = 5, and angle A = 60°. Find BC. The known angle is between sides AB and AC, so use the cosine rule with a = BC: BC^2 = 7^2 + 5^2 − 2(7)(5)cos 60° = 49 + 25 − 70(0.5) = 39. Thus BC = sqrt(39) ≈ 6.24. The area is (1/2)(7)(5)sin 60° = 35sqrt(3)/4 ≈ 15.2.
Traps
Using the sine rule when the included angle is known; the ambiguous sine-rule case; calculator in the wrong mode; rounding before the final step; treating a vector as a scalar; forgetting that a·b < 0 means an obtuse angle.
Lines and planes
A line r=a+tb requires a nonzero direction vector. A plane with normal n through a has equation n·(r-a)=0. Substitute the line into the plane equation to find an intersection parameter, then substitute that parameter back to obtain the actual point. A zero dot product with the normal makes a direction parallel to the plane; it does not by itself prove that the line lies in the plane. Test a point as well. For a nonzero direction b, the perpendicular distance between two points on its line follows from the length of their difference vector.