Understand and use the definitions of secant, cosecant and cotangent and of arcsin, arccos and arctan; their relationships to sine, cosine and tangent; understanding of their graphs; their ranges and domains

This topic introduces the reciprocal trigonometric functions: secant (sec), cosecant (cosec), and cotangent (cot), along with their inverse functions: arcsin, arccos, and arctan. These functions extend the standard sine, cosine, and tangent, allowing you to solve a wider range of trigonometric equations and model more complex periodic phenomena. Understanding their definitions, graphs, domains, and ranges is essential for A-Level Mathematics, particularly in calculus (differentiation and integration) and solving trigonometric equations.

The reciprocal functions are defined as secθ = 1/cosθ, cosecθ = 1/sinθ, and cotθ = 1/tanθ = cosθ/sinθ. Their graphs have vertical asymptotes where the original function is zero, and they exhibit periodic behaviour with distinct ranges. The inverse functions (arcsin, arccos, arctan) are used to find angles given a trigonometric ratio, but they are restricted to principal values to ensure they are functions. For example, arcsin has domain [-1,1] and range [-π/2, π/2], while arccos has range [0, π] and arctan has range (-π/2, π/2).

Mastering these functions is crucial for solving equations like secθ = 2 or arccos(x) = π/3, and for integrating expressions involving 1/(1+x²) or 1/√(1-x²). They also appear in modelling real-world contexts such as alternating current circuits (secant) and projectile motion (arctan). A solid grasp of these concepts will prepare you for more advanced topics in pure mathematics and applied modules.