Trigonometry

2026 Syllabus Objectives

By the end of this topic, you should be able to:

  1. Understand the relationship of the secant, cosecant and cotangent functions to cosine, sine and tangent, and use properties and graphs of all six trigonometric functions for angles of any magnitude

  2. Use trigonometrical identities for the simplification and exact evaluation of expressions, and in the course of solving equations, including:

    • sec²θ = 1 + tan²θ and cosec²θ = 1 + cot²θ
    • The expansions of sin(A ± B), cos(A ± B) and tan(A ± B)
    • The formulae for sin 2A, cos 2A and tan 2A
    • The expression of a sin θ + b cos θ in the forms R sin(θ ± α) and R cos(θ ± α)
    • Simplifying expressions like cos(x – 30°) – 3 sin(x – 60°)
    • Solving equations like tan θ + cot θ = 4, 2 sec²θ - tan θ = 5, 3 cos θ + 2 sin θ = 1

1. Reciprocal Trigonometric Functions

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What are secant, cosecant and cotangent?

In addition to sine, cosine and tangent, there are three more trigonometric functions. These are called reciprocal functions because they are simply the "flipped" versions of the original three.

The three reciprocal functions are:

  • Secant (sec) is the reciprocal of cosine:

    sec θ = 1/cos θ

  • Cosecant (cosec) is the reciprocal of sine:

    cosec θ = 1/sin θ

  • Cotangent (cot) is the reciprocal of tangent:

    cot θ = 1/tan θ

Understanding reciprocals

A reciprocal simply means "one divided by that number". For example:

  • The reciprocal of 2 is 1/2
  • The reciprocal of 5 is 1/5
  • The reciprocal of cos θ is 1/cos θ (which we call sec θ)

Alternative ways to write cotangent

Since tan θ = sin θ/cos θ, we can also write cotangent as:

cot θ = 1/tan θ = cos θ/sin θ

This gives us two useful ways to write cot θ.

Important points to remember

  • When the denominator is zero, the function is undefined. For example:

    • sec θ is undefined when cos θ = 0 (at θ = 90°, 270°, etc.)
    • cosec θ is undefined when sin θ = 0 (at θ = 0°, 180°, 360°, etc.)
    • cot θ is undefined when sin θ = 0

  • These functions work for angles of any size – positive, negative, or greater than 360°.

Graphs of the six trigonometric functions

You should be familiar with the shapes and key features of all six trig function graphs:

sin θ and cosec θ:

  • sin θ oscillates between -1 and +1
  • cosec θ = 1/sin θ has vertical asymptotes (vertical lines the graph never touches) wherever sin θ = 0
  • cosec θ has no maximum or minimum values – it goes to infinity

cos θ and sec θ:

  • cos θ oscillates between -1 and +1
  • sec θ = 1/cos θ has vertical asymptotes wherever cos θ = 0
  • sec θ has no maximum or minimum values – it goes to infinity

tan θ and cot θ:

  • tan θ repeats every 180° and has vertical asymptotes at odd multiples of 90° (90°, 270°, etc.)
  • cot θ repeats every 180° and has vertical asymptotes at multiples of 180° (0°, 180°, 360°, etc.)
  • Both functions can take any value from negative infinity to positive infinity

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