Geometry and Trigonometry

Shared with Applications

Both guides state 3.1 to 3.3 as content common to the two courses, so these are the same pages Applications students use rather than a second copy that could drift.

3.1 · Three-dimensional geometry SHARED Distances and angles in solids, with Pythagoras twice. The angle is not to the nearest edge, it is to the base diagonal. 3.2 · The sine and cosine rules SHARED Right-angled ratios, both rules, and the area of a triangle. The sine rule hands back the wrong angle and the angle sum is what tells you. 3.3 · Applications, including bearings SHARED Trigonometry in context, with bearings, elevation and depression. The angle at the turn is neither bearing, and the wrong one lands 1.04 km short.

Analysis and Approaches, Standard Level

Radians are Standard Level on this course, which is the first real divergence from Applications, where they wait until Higher Level. Everything from here is in radians unless a degree symbol appears.

3.4 · The circle in radians Radian measure, arc length, sector area, and the exact forms Paper 1 wants. The sector is 18.85 and the segment inside it is 3.26. 3.5 · The unit circle and exact values Cosine and sine as coordinates, tan as their ratio, the exact ratios and their multiples, and the ambiguous case. The sine stays at 0.866 and the cosine flips to −0.5. 3.6 · The identities The Pythagorean identity, the double angle identities for sine and cosine, and the relationships between the ratios. sin 2x is 0.9922. Doubling the sine gives 1.3229, which is not a sine. 3.7 · The circular functions Amplitude, period and principal axis, and a sin(b(x + c)) + d as a transformation. The period is 2.094, and the 2π answer is three times too long. 3.8 · Trigonometric equations Solving in a given interval, graphically and analytically, including the ones that become quadratics. Two solutions on 0 to 2π and four on 0 to 4π. Same equation.

Higher Level only

Ten more sub-topics. The first three finish the trigonometry, and the remaining seven build vectors from the concept up to planes, which is where the topic is heading all along.

3.9 · Reciprocal ratios and inverse functions HL Secant, cosecant and cotangent, the two further Pythagorean identities, and arcsin, arccos and arctan with their domains. arcsin 0.5 is 0.5236 and 1/sin 0.5 is 2.0858. 3.10 · Compound angle identities HL The addition formulae, and the double angle identity for tangent derived from them. sin 45 + sin 45 is 1.4142, which is not a sine at all. 3.11 · Symmetry of the graphs HL How the three functions relate to each other, read off the symmetry of their graphs. Adding π flips the sine and cosine and leaves the tangent alone. 3.12 · Vectors HL The concept, position and displacement vectors, base vectors, components, scalar multiples, magnitude, and proofs. Two vectors of length 5 can add to 7.0711, or to nothing. 3.13 · The scalar product HL Its definition and properties, the angle between two vectors, and the tests for perpendicular and parallel. Two vectors of length 5 with a product of exactly zero. 3.14 · The vector equation of a line HL r = a + λb, the parametric and Cartesian forms, the angle between two lines, and kinematics. A line has infinitely many equations, so your answer and the book's can both be right. 3.15 · Coincident, parallel, intersecting and skew HL Telling the four cases apart, and finding a point of intersection when there is one. The plan crosses at (1, 2) and the heights there are 3 and 2. 3.16 · The vector product HL Its definition and properties, and what its magnitude measures. 15 is the parallelogram and 7.5 is the triangle. 3.17 · The equation of a plane HL All three forms: the parametric one, r dotted with n, and ax + by + cz = d. The numbers in front of x, y and z point out of the plane, not along it. 3.18 · Intersections and angles HL A line with a plane, two planes, three planes, and the angles between them. 63.61° to the normal is 26.39° to the plane.

Doing Applications instead?

Sub-topics 3.1 to 3.3 above are yours too, word for word, because both guides state them as common content. Applications then continues with Voronoi diagrams and graph theory rather than planes. Applications Topic 3 is here, all sixteen sub-topics.