E: Trigonometry: 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.

The idea

This topic extends trigonometry beyond sine, cosine and tangent. Define the reciprocals: sec θ = 1/cos θ, cosec θ = 1/sin θ, and cot θ = cos θ/sin θ (equivalently 1/tan θ where tan θ is defined and non-zero). The inverse functions arcsin, arccos and arctan return angles for given ratios. Graphs, domains and ranges are essential. Sec and cosec have vertical asymptotes where cos or sin are zero, with ranges |y| ≥ 1; cot has asymptotes where sin is zero and range all real numbers. The inverse graphs are the reflections of the restricted sine, cosine and tangent graphs in y = x: arcsin has domain [-1, 1] and range [-π/2, π/2]; arccos has domain [-1, 1] and range [0, π]; arctan has domain all real numbers and range (-π/2, π/2). Use these relationships to solve equations and prove identities.

What earns the marks

• Defines sec θ, cosec θ and cot θ correctly as reciprocals of cos θ, sin θ and tan θ, including cot θ = cos θ/sin θ and noting where tan θ is defined and non-zero.
• States the domain and range of arcsin, arccos and arctan, including the restricted ranges that make them functions.
• Sketches or describes the graphs of sec, cosec and cot, identifying vertical asymptotes and key features such as minimum points.
• Sketches or describes the graphs of arcsin, arccos and arctan, identifying their domains, ranges and key features such as endpoints and the horizontal asymptotes of arctan.
• Uses the relationships between trigonometric functions and their reciprocals to simplify expressions or solve equations.
• Applies inverse trigonometric functions correctly to find angles in a given range, showing awareness of principal values.

Common mistakes and how to put them right

• Confusing the notation: writing sec θ as the inverse function (cos⁻¹ θ) rather than the reciprocal (1/cos θ).

  Put it right: Sec θ = 1/cos θ, while cos⁻¹ θ = arccos θ.

• Assuming the range of arcsin is all real numbers or forgetting the restricted range.

  Put it right: Arcsin x is defined only for x in [-1, 1] and returns values in [-π/2, π/2].

• Drawing the graph of sec θ as a continuous curve without asymptotes.

  Put it right: Sec θ has vertical asymptotes at θ = π/2 + nπ, where cos θ = 0.

• Defining cot θ as 1/tan θ without qualification.

  Put it right: Cot θ = cos θ/sin θ, which is defined at θ = π/2 where tan θ is undefined but cot θ = 0.

• Treating arctan as having a bounded domain or drawing it with vertical asymptotes.

  Put it right: Arctan has domain all real numbers and horizontal asymptotes y = ±π/2.

Exam tips

• When solving equations involving sec, cosec or cot, first rewrite them in terms of sin, cos or tan to avoid algebraic errors.
• For inverse trigonometric functions, always consider the principal value range and check whether additional solutions exist in the required interval.
• Sketch the reciprocal graphs by first sketching the original sine, cosine or tangent graph, then reciprocating the y-values: where the original is 0, draw a vertical asymptote; where the original is 1 or -1, the reciprocal graph touches it; where the original is large, the reciprocal is close to 0.
• For inverse graphs, reflect the restricted original graph in y = x; mark endpoints for arcsin and arccos and horizontal asymptotes for arctan.