Chapter 5: Trigonometry, Coordinate Geometry, Vectors, and Complex Numbers

Trigonometry

Angles are measured in degrees or radians; radians . Arc length and sector area (with in radians).

Fundamental identities: , , .

Compound angles:

Double angles: ; ; .

The R-form: with and . It gives the maximum value and is widely used for solving equations and in alternating current and wave problems.

General solutions (with ): ; ; .

Triangles: sine rule ; cosine rule ; area .

Worked example. Solve . Here and , so . Then , so and or , giving or in .

Coordinate geometry

Distance: . Midpoint: . Section formula (ratio ): .

Straight line: ; general form . Parallel lines have equal gradients; perpendicular lines satisfy . The angle between lines satisfies .

Perpendicular distance from to : .

Circle: centre , radius : . In general form the centre is and the radius is . A line is tangent to a circle when the distance from the centre to the line equals the radius.

Conics: the parabola has focus and directrix ; the ellipse and the hyperbola are studied to the extent the syllabus requires.

Vectors

A vector has magnitude and direction. In components, . The scalar (dot) product is ; it is zero for perpendicular vectors. The vector (cross) product has magnitude and direction given by the right-hand rule; it gives areas and torques. Vector equation of a line: . Vector methods prove geometric results (for example, that the diagonals of a parallelogram bisect each other) and describe forces, velocities, and fields in physics.

Complex numbers

A complex number is with ; , . The conjugate is and . In polar form, with and (Euler’s formula). Multiplication multiplies moduli and adds arguments. De Moivre’s theorem:

The th roots of are for , equally spaced on a circle in the Argand diagram.

Worked example. : , . So .

Matrices and determinants

For , , and when , . A system has the unique solution when is invertible.

Common mistakes

Using degrees where radians are required in calculus.

Losing solutions by dividing by a trigonometric expression that can be zero.

Confusing with .

Forgetting the argument’s quadrant for complex numbers.

Practice questions

Show that .

Find the distance from to the line . [13/5 = 2.6]

Find the centre and radius of . [(3, −2), 5]

Find the cube roots of unity.