Trigonometry & Vectors

This module brings together circle-based trigonometry and introductory vector algebra, preparing learners for mechanics and later calculus. Standard angles, identities and equation-solving techniques are revisited with precision, while vectors are developed from geometric intuition to the dot and cross products. Angular measure, magnitude and direction are treated carefully. By its end, learners can reason fluently about rotations, triangles and vector relationships in two and three dimensions.

Learning Outcomes

Convert between degree and radian measure and compute arc length and sector area (apply)

Evaluate trigonometric ratios of standard angles exactly (apply)

Prove and apply Pythagorean, double-angle and sum-and-difference identities (apply)

Solve trigonometric equations in a stated interval (apply)

Compute vector magnitude, direction, dot and cross products (apply)

Analyse geometric situations using the sine and cosine rules (analyse)

Evaluate when a given vector argument is valid in three dimensions (evaluate)

Construct and interpret vector proofs of geometric statements (create)

Topics Covered

Angles, radians and the unit circle

Exact values of sin, cos, tan at standard angles

Graphs of sin, cos, tan and their transformations

Pythagorean, reciprocal and quotient identities

Sum-to-product and double-angle identities

Trigonometric equations and general solutions

The sine and cosine rules

Area of a triangle using trigonometry

Vector notation and components

Vector magnitude and unit vectors

Vector addition and scalar multiplication

Dot product and the angle between vectors

Cross product and area/volume applications

Direction cosines and three-dimensional vectors

Applications to geometry and mechanics

Skills Developed

Exact value recall for standard angles Manipulating trigonometric identities Solving trigonometric equations in an interval Working with vectors in 2D and 3D Dot and cross-product computation Applying sine/cosine rules Translating geometric problems into vector form Interpreting magnitude and direction

Suggested Resources

STRAIT Trigonometry & Vectors workbook (TutorDA, 2025)

Stewart, J. Precalculus (7th ed.), chapters 5-8

Khan Academy: Trigonometry and Vectors playlists

Paul's Online Math Notes: Trig section

GeoGebra 3D vector grapher

Desmos unit-circle demonstration

TutorDA practice bank: A-M3 drill sets

Assessment

Assessment emphasises both the symbolic skill of manipulating identities and the geometric judgement needed for vector problems, split across an exam and short-form class tests.

Written examination (2 hr)60%

Class tests and coursework40%

Practice bank: 198 MCQs · Timed mock & self-paced modes

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