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3 Geometry and trigonometry

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  1. 3.1 Geometry and trigonometry - SL content

    1. • Use distance and midpoint in three-dimensional space. • Use volume and surface area of common 3D solids and combinations. • Identify right-angled triangles in 3D objects to find lengths and angles.

    2. • Use sine, cosine and tangent in right-angled triangles. • Use sine rule, cosine rule and area formula 1/2 ab sin C. • Ambiguous sine-rule case is not included here.

    3. • Apply right and non-right triangle trigonometry, including Pythagoras' theorem. • Use angles of elevation/depression and bearings; construct labelled diagrams from text.

    4. • Use radians for angles. • Find arc length and sector area; express radians exactly with pi or as decimals.

    5. • Define sin theta and cos theta using the unit circle; tan theta=sin theta/cos theta. • Use quadrant relationships and exact trig values for standard angles. • Include ambiguous sine-rule case.

    6. • Use cos^2 theta + sin^2 theta = 1. • Use double-angle identities for sine and cosine. • Use trig-ratio relationships without necessarily finding the angle.

    7. • Graph sin x, cos x and tan x with amplitude and periodicity. • Use transformations of a sin(b(x+c))+d in degrees or radians. • Apply to contexts such as tides and Ferris wheels.

    8. • Solve trigonometric equations on finite intervals graphically and analytically. • Include equations reducible to quadratics in sin x, cos x or tan x. • General trig solutions are not required.

  2. 3.2 Geometry and trigonometry - AHL content

    1. AHL 3.9 (HL)—Reciprocal and inverse trigonometric functions

      • Define sec theta, cosec theta and cot theta. • Use identities 1+tan^2 theta=sec^2 theta and 1+cot^2 theta=cosec^2 theta. • Know domains, ranges and graphs of arcsin x, arccos x and arctan x.

    2. AHL 3.10 (HL)—Compound-angle identities

      • Use compound-angle identities and the double-angle identity for tan. • Derive double-angle identities from compound-angle identities.

    3. AHL 3.11 (HL)—Further trigonometric equations

      • Solve more complex trigonometric equations in finite intervals. • Use reciprocal, inverse and compound-angle identities where appropriate.

    4. AHL 3.12 (HL)—Vectors

      • Use position vectors, displacement vectors and directed line segment representation. • Use base vectors i, j, k; components, magnitude, unit vectors and vector algebra. • Prove geometric properties using vectors.

    5. AHL 3.13 (HL)—Scalar product

      • Use scalar product, angle between vectors, perpendicular and parallel vector conditions. • Apply v dot w = |v||w| cos theta and scalar product properties.

    6. AHL 3.14 (HL)—Vector equations of lines

      • Use vector, parametric and Cartesian equations of lines in 2D and 3D. • Find angles between lines using scalar product of direction vectors. • Interpret line parameter as time in simple kinematics.

    7. AHL 3.15 (HL)—Line relationships in 3D

      • Distinguish coincident, parallel, intersecting and skew lines. • Find points of intersection where they exist.

    8. AHL 3.16 (HL)—Vector product

      • Use vector/cross product v x w = |v||w| sin theta n. • Know properties, right-hand rule direction and parallel-vector condition. • Use |v x w| for parallelogram and triangle areas.

    9. AHL 3.17 (HL)—Planes

      • Use vector equations of a plane r=a+lambda b+mu c and r dot n=a dot n. • Use Cartesian equation ax+by+cz=d.

    10. AHL 3.18 (HL)—Intersections and angles in 3D

      • Find intersections of line-plane, two-plane and three-plane systems. • Find angles between a line and a plane, and between two planes. • Interpret solutions geometrically.