Edexcel IAL - Pure Maths 3- 2.1 Secant, cosecant and cotangent, and of arcsin, arccos and arctan Functions- Study notes  - New syllabus

Secant, Cosecant, Cotangent and Inverse Trigonometric Functions

In addition to sine, cosine and tangent, there are three related trigonometric functions:

• Secant
• Cosecant
• Cotangent

There are also the inverse trigonometric functions:

• \( \arcsin x \)
• \( \arccos x \)
• \( \arctan x \)

Secant, Cosecant and Cotangent

These functions are defined in terms of sine, cosine and tangent.

Function	Definition	Undefined when	Graph
Secant	\( \sec x = \dfrac{1}{\cos x} \)	\( \cos x = 0 \)
Cosecant	\( \csc x = \dfrac{1}{\sin x} \)	\( \sin x = 0 \)
Cotangent	\( \cot x = \dfrac{1}{\tan x} = \dfrac{\cos x}{\sin x} \)	\( \sin x = 0 \)

Relationships Between Trigonometric Functions 

• \( \sin x = \dfrac{1}{\csc x} \)
• \( \cos x = \dfrac{1}{\sec x} \)
• \( \tan x = \dfrac{1}{\cot x} \)

Inverse Trigonometric Functions

Inverse trigonometric functions reverse the effect of sine, cosine and tangent.

They are written as:

\( \arcsin x,\ \arccos x,\ \arctan x \)

These are functions only if their domains are restricted.

Restricted Domains and Ranges

Function	Domain	Range	Graph
\( \arcsin x \)	\( -1 \le x \le 1 \)	\( -\dfrac{\pi}{2} \le y \le \dfrac{\pi}{2} \)
\( \arccos x \)	\( -1 \le x \le 1 \)	\( 0 \le y \le \pi \)
\( \arctan x \)	\( x \in \mathbb{R} \)	\( -\dfrac{\pi}{2} < y < \dfrac{\pi}{2} \)

Angles in Degrees and Radians

Angles may be measured in:

• Degrees \( (^\circ) \)
• Radians

Conversion formula:

\( 180^\circ = \pi \text{ radians} \)

Examples:

• \( 30^\circ = \dfrac{\pi}{6} \)
• \( \dfrac{\pi}{3} = 60^\circ \)