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 $t = \pi$.