Topic 3: Geometry and Trigonometry
SL 3.1
Mensuration
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SL 3.2
SL 3.3
Sine Rule and Cosine Rule
Applications of Trigonometry
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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.
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Use of sine, cosine and tangent ratios to find the sides and
angles of right-angled triangles.
𝑎
𝑏
𝑐
The sine rule: 𝑠𝑖𝑛 𝐴 = 𝑠𝑖𝑛 𝐵 = 𝑠𝑖𝑛 𝐶 .
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The cosine rule: 𝑐 = 𝑎 + 𝑏 − 2𝑎𝑏 𝑐𝑜𝑠 𝐶; 𝑐𝑜𝑠 𝐶 =
2
2
1
2
2
2
2
2
𝑎 +𝑏 −𝑐
2𝑎𝑏
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Area of a triangle as
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Applications of right and non-right angled trigonometry,
including Pythagoras’ theorem.
Angles of elevation and depression.
Construction of labelled diagrams from written statements.
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.
𝑎𝑏 𝑠𝑖𝑛 𝐶.
SL 3.4
The Unit Circle
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The circle: radian measure of angles; length of an arc; area of
a sector.
SL 3.5
Trigonometry with Radian
Measurement
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Definition of 𝑐𝑜𝑠 θ, 𝑠𝑖𝑛 θ in terms of the unit circle.
𝑠𝑖𝑛 θ
Definition of 𝑡𝑎𝑛 θ as 𝑐𝑜𝑠 θ .
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Exact values of trigonometric ratios of
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multiples.
Extension of the sine rule to the ambiguous case.
π
6
,
π
4
,
π
3
,
π
2
and their
SL 3.6
Trigonometric Identities
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SL 3.7
Trigonometric Functions and
Transformations
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SL 3.8
Trigonometric Equations
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AHL 3.9
Reciprocal Trigonometric Ratios
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2
2
The Pythagorean identity 𝑐𝑜𝑠 θ + 𝑠𝑖𝑛 θ = 1. Double angle
identities for sine and cosine.
The relationship between trigonometric ratios.
The circular functions 𝑠𝑖𝑛 𝑥, 𝑐𝑜𝑠 𝑥, and 𝑡𝑎𝑛 𝑥; amplitude, their
periodic nature, and their graphs.
Composite functions of the form 𝑓(𝑥) = 𝑎 𝑠𝑖𝑛 (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 𝑠𝑖𝑛 𝑥, 𝑐𝑜𝑠 𝑥 or
𝑡𝑎𝑛 𝑥.
Definition of the reciprocal trigonometric ratios 𝑠𝑒𝑐 θ, 𝑐𝑠𝑐 θ
and 𝑐𝑜𝑡 θ.
Pythagorean identities:
2
2
2
2
○
1 + 𝑡𝑎𝑛 θ = 𝑠𝑒𝑐 θ
○
○
1 + 𝑐𝑜𝑡 θ = 𝑐𝑠𝑐 θ
The inverse functions 𝑓(𝑥) = 𝑎𝑟𝑐𝑠𝑖𝑛 𝑥,
𝑓(𝑥) = 𝑎𝑟𝑐𝑐𝑜𝑠 𝑥, 𝑓(𝑥) = 𝑎𝑟𝑐𝑡𝑎𝑛 𝑥; their domains and
ranges; their graphs.
AHL 3.10
Compound Angle Identities
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Compound angle identities.
Double angle identity for 𝑡𝑎𝑛.
AHL 3.11
Symmetric Properties of
Trigonometric Function
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Relationships between trigonometric functions and the
symmetry properties of their graphs.
AHL 3.12
Vectors
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Concept of a vector; position vectors; displacement vectors.
Representation of vectors using directed line segments.
Base vectors 𝑖, 𝑗, 𝑘.
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Components of a vector: 𝑣 =< 𝑣1, 𝑣2, 𝑣3 >= 𝑣1𝑖 + 𝑣2𝑗 + 𝑣3𝑘.
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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.
𝑣
○ magnitude of a vector, |𝑣|; unit vectors, |𝑣| .
○
→
→
position vectors 𝑂𝐴 = 𝑎, 𝑂𝐵 = 𝑏.
→
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○ displacement vector 𝐴𝐵 =− 𝑏𝑎.
Proofs of geometrical properties using vectors.
AHL 3.13
Relationships between Vectors
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The definition of the scalar product of two vectors.
The angle between two vectors.
Perpendicular vectors; parallel vectors.
AHL 3.14
Vector Equation
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Vector equation of a line in two and three dimensions:
𝑟 = 𝑎 + λ𝑏.
The angle between two lines.
Simple applications to kinematics.
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AHL 3.15
Linear Relationships in 3D Space
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Coincident, parallel, intersecting and skew lines,
distinguishing between these cases.
Points of intersection.
AHL 3.16
Vector Products
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The definition of the vector product of two vectors.
Properties of the vector product.
Geometric interpretation of |𝑣 × 𝑤|.
AHL 3.17
Vectors in the Plane
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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.
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AHL 3.18
Intersections between Lines and
Planes
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Intersections of: a line with a plane; two planes; three planes.
Angle between: a line and a plane; two planes.