These are the statements describing what students need to learn:
[Higher Level only statements are in blue]
- the distance between two points in three-dimensional space, and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane
- use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles. use of the sine rule, cosine rule and the sine formula for finding the area of a triangle
- applications of right and non-right angled trigonometry, including Pythagoras’s theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements
- the circle: radian measure of angles; length of an arc; area of a sector
- Definition of cosθ, sinθ in terms of the unit circle. Definition of tanθ as sinθ ÷ cosθ. Exact values of trigonometric ratios of θ, π/6, π/4, π/3, π/2 and their multiples. Extension of the sine rule to the ambiguous case.
- the Pythagorean identity cos2θ+sin2θ=1. Double angle identities for sine and cosine. The relationship between trigonometric ratios
- the circular functions sinx, cosx, and tanx; amplitude, their periodic nature, and their graphs. Composite functions of the form f(x)=asin(b(x+c))+d. Transformations. Real-life contexts
- solving trigonometric equations in a finite interval, both graphically and analytically. Equations leading to quadratic equations in sinx, cosx or tanx
- Definition of the reciprocal trigonometric ratios secθ, cosecθ and cotθ.
Pythagorean identities:
1+tan2θ=sec2θ
1+cot2θ=cosec2θ
The inverse functions f(x)=arcsinx,
f(x)=arccosx,f(x)=arctanx; their domains and ranges; their graphs. - Compound angle identities. Double angle identity for tan.
- Relationships between trigonometric functions and the symmetry properties of their graphs.
- Concept of a vector; position vectors; displacement vectors.
Representation of vectors using directed line segments.
Base vectors i, j, k.
Components of a vector:
v=v1i+v2j+v3k.
Algebraic and geometric approaches to the following:
the sum and difference of two vectors
the zero vector 0, the vector -v
multiplication by a scalar, kv, parallel vectors
magnitude of a vector, |v|; unit vectors, v÷|v|
position vectors OA=a, OB=b
displacement vector AB=b-a
Proofs of geometrical properties using vectors.
- The definition of the scalar product of two vectors. The angle between two vectors. Perpendicular vectors; parallel vectors.
- Vector equation of a line in two and three dimensions:
r=a+λb.
The angle between two lines.
Simple applications to kinematics. - Coincident, parallel, intersecting and skew lines, distinguishing between these cases. Points of intersection.
- The definition of the vector product of two vectors.
Properties of the vector product.
Geometric interpretation of |v×w| - Vector equations of a plane:
r=a+λb+μc, where b and c are non-parallel vectors within the plane.
r·n=a·n, where n is a normal to the plane and a is the position vector of a point on the plane.
Cartesian equation of a plane ax+by+cz=d. - Intersections of: a line with a plane; two planes; three planes.
Angle between: a line and a plane; two planes.
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