# 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.