Secant, Cosecant and Cotangent

Introduction to Secant, Cosecant and Cotangent

Basic Concepts

  • Understand the definition and function of the secant (sec), cosecant (csc) and cotangent (cot) trigonometric ratios. Both secant and cosecant are reciprocals of cosine and sine functions respectively, while cotangent is the reciprocal of tangent function.
  • Familiarise yourself with how these functions are derived from the unit circle and the right-angle triangle definition of trigonometric ratios.

Function Representation

  • Advise that for any given angle θ,
    • sec θ = 1/cos θ
    • csc θ = 1/sin θ
    • cot θ = 1/tan θ or cot θ = cos θ / sin θ
  • Be aware that these functions are undefined wherever the denominators, i.e., sin θ or cos θ, are zero because division by zero is undefined in mathematics.

Properties of Secant, Cosecant and Cotangent

Identities and Relationships

  • Remember the importance of Pythagorean identities involving secant and cosecant — sec² θ - 1 = tan² θ and csc² θ - 1 = cot² θ. These identities establish vital relationships among trigonometric functions.
  • Understand the identities involving mixed ratios, such as cot θ = cos θ / sin θ, which often simplify trigonometric expressions or equations.

Angles and Trigonometric Values

  • Memorise the function values of sec, csc, and cot for common angles, such as 0, π/6, π/4, π/3, π/2, etc. For sec and csc, recognise when the function values are undefined.

Graphing Secant, Cosecant and Cotangent

Graphical Representation

  • Be capable of graphing these functions precisely, acknowledging that they exhibit periodic behaviour similar to sine, cosine, and tangent functions.
  • Understand that sec and csc graphs include vertical asymptotes where the functions are undefined, and cot graph contains discontinuities at multiples of π where it is undefined.

Amplitude, Period and Phase Shift

  • Accept that these straight functions do not have an amplitude; instead, they tend to infinity.
  • Comprehend that the period of sec and csc is 2π, just like for cosine and sine, while for cot, it is π, half the period of tangent.
  • Get familiar with identifying any phase shifts or vertical shifts from the standard function graphs.

Real World Applications

  • Appreciate the employment of secant, cosecant, and cotangent in various fields of science and engineering, offering additional viewpoint in problems involving wave motion, resonance, oscillations, electrical networks, etc.
  • Acknowledge their critical role in integral calculus, specifically in solvability and simplification of complex integrals.

Problems and Solutions

  • Practice solving trigonometric equations involving sec, csc, and cot. Recognise how factoring, method of substitution, utilisation of identities, etc., are employed.
  • Solve problems involving the use of secant, cosecant, and cotangent in physics and engineering.