The trigonometry extends beyond sine, cosine, and tangent to include their reciprocals: secant, cosecant, and cotangent. This section explores their definitions, properties, and graphical representations.

## Understanding Extended Trigonometric Functions

### Secant (sec)

**Definition: **The secant of an angle in a right-angled triangle is the reciprocal of the cosine, $(\text{sec}(t) = \frac{1}{\cos(t)}$.

**Characteristics:**

- Undefined where cosine is zero (at odd multiples of $\frac{\pi}{2} )$.
- Range:

- An even function: $\text{sec}(-t) = \text{sec}(t)$

Image courtesy of Online Math Learning

### Cosecant (csc)

**Definition: **The cosecant is the reciprocal of sine, $\text{csc}(t) = \frac{1}{\sin(t)}$.

**Characteristics:**

- Undefined where sine is zero (at integer multiples of $\pi )$.
- Shares secant's range.
- An even function.

Image courtesy of Online Math Learning

### Cotangent (cot)

**Definition: **Cotangent is the reciprocal of tangent,

$\text{cot}(t) = \frac{1}{\tan(t)}$ or $\text{cot}(t) = \frac{\cos(t)}{\sin(t)}$

**Characteristics:**

- Undefined where sine is zero.
- An odd function:

- .Range: All real numbers.

Image courtesy of Wikipedia

## Application Across Angles of Any Magnitude

**Example 1: Understanding Asymptotes**

**Problem: **$\text{sec}(270^\circ)$.

**Solution: **

Since $\cos(270^\circ) = 0 , \text{sec}(270^\circ)$ is undefined, indicating a vertical asymptote on the secant graph.

**Example 2: Graphical Analysis**

**Problem:** Analyse $\text{cot}(t)$ for $t$ in $[0, 2\pi]$.

**Solution: **

The cotangent graph shows periodicity and asymptotes at points where $\sin(t) = 0$.

**Example 3: Graph Interpretation**

**Problem:** Interpret the $\text{csc}(t)$ graph for $t$ in $[0, 2\pi]$.

**Solution:**

The cosecant graph displays inverted U-shaped curves, undefined at $t = 0$ and $t = \pi$.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.