IB Maths AA 3.8 Notes

This page contains our IB Maths AA notes for 3.8. By reading each one of these notes, you will fully cover the content for IB Maths AA 'Advanced trigonometry'.

All Topic 3 Sub-topics 3.1: Fundamentals Of Geometry 3.2: Introduction To Trigonometry 3.3: Applied Trigonometry 3.4: Circles 3.5: Unit Circle 3.6: Trigonometric Identities 3.7: Trigonometric Graphs (HL)3.8: Advanced trigonometry (HL)3.9: Compound angles (HL)3.10: Vectors (HL)3.11: Lines (HL)3.12: Planes

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Chapters

1. Reciprocal trigonometric functions

2. Reciprocal Pythogorean identities

3. Inverse trigonometric functions

4. Symmetry

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Reciprocal trigonometric functions

The reciprocal trigonometric functions are defined by:

$\sec x=\frac{1}{\cos x}$
$\cosec x=\frac{1}{\sin x}$
$\cot x=\frac{1}{\tan x}=\frac{\cos x}{\sin x}$

These are useful because they appear naturally when rearranging identities and solving equations.

Since they are reciprocals:

$\sec x$ is undefined when $\cos x=0$
$\cosec x$ is undefined when $\sin x=0$
$\cot x$ is undefined when $\tan x=0$ , equivalently when $\sin x=0$

We can see this below:

Further to this, reciprocal functions have a useful graphical relationship with $\sin x$ , $\cos x$ and $\tan x$ .

For a reciprocal function:

it equals $1$ where the original function equals $1$
it equals $-1$ where the original function equals $-1$
it becomes very large positive or negative where the original function approaches $0$
zeros of the original function become vertical asymptotes of the reciprocal graph

Thus, with the reciprocal trigonometric functions:

$y=\sec x$ has vertical asymptotes where $\cos x=0$
$y=\cosec x$ has vertical asymptotes where $\sin x=0$
$y=\cot x$ has vertical asymptotes where $\tan x=0$

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