VCE Specialist Mathematics/Units 3 and 4: Specialist Mathematics/Circular Functions

Preface

Formal Definition: In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. They are used to relate the angles of a triangle to the lengths of the sides of a triangle.

Translation: Understanding the various functions that can be applied to and taken from (graphs) the unit circle, and includes recognizing and using the various algebraic identities to manipulate trigonometric functions and equations.

Graphing Functions

Sin

General formula:

$y=a\sin(n(x-b))+c$

General Notes:

General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".

A period is equal to $[{\frac {2\pi }{n}}]$

The domain, unless restricted, is $x\in \mathbb {R}$

The range is equal to $[\pm a+c]$ , as the range of $y=\sin(x),y\in [-1,1]$ , see unit circle.

The horizontal translation of $b$ is reflected in the x-intercepts.

Easily sketched by applying graphic transformations, from VCE Mathematical Methods

Cos

General formula:

$y=a\cos(n(x-b))+c$

General Notes:

General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".

The domain, unless restricted, is $x\in \mathbb {R}$ , as $y=\cos(x),x\in \mathbb {R}$

A period is equal to $[{\frac {2\pi }{n}}]$ , as the factor of n

The range is equal to $[\pm a+c]$ , as the range of $y=\cos(x),y\in [-1,1]$ , see unit circle.

The horizontal translation of $b$ is reflected in the x-intercepts.

Easily sketched by applying graphic transformations, from VCE Mathematical Methods

Tan

General formula:

$y=a\tan(n(x-b))+c$

General Notes:

General formula achieved isolating the coefficient of x (n), i.e. the "lonely x rule".

A period is equal to $[{\frac {\pi }{n}}]$

The domain, $x\in \mathbb {R} \setminus {\frac {k\pi }{2n}},k\in \mathbb {N}$ , as $y=\tan(x),x\in \mathbb {R} \setminus {\frac {k\pi }{2}},k\in \mathbb {N}$ , indicating the asymptotes.

The range, unless restricted, is $y\in \mathbb {R}$ , as the range of $y=\tan(x),y\in \mathbb {R}$ , see unit circle.

The horizontal translation of $b$ is reflected in the x-intercepts.

Easily sketched by applying graphic transformations, from VCE Mathematical Methods

Arcsin

Also known as $Sin^{-}1$ or $sin^{-}$

Arccos

Also known as $Cos^{-}1$ or $cos^{-}$

Arctan

Also known as $Tan^{-}1$ or $tan^{-}$

Examples

Graphing Functions

General Method

Trigonometric Functions Method

Reciprocal Trigonometric Functions Method

Inverse Trigonometric Functions Method

Sin

Cos

Tan

Arcsin

Arccos

Arctan

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