# VCE Specialist Mathematics/Units 3 and 4: Specialist Mathematics/Formulae

 « VCE Specialist Mathematics Formulae » Mechanics Practice SACS

## Preface

This is a list of all formulae needed for Units 3 and 4: Specialist Mathematics.

## Formulae

### Ellipses, Circles and Hyperbolas

#### Ellipses

General formula:

• $\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1$

General Notes:

• Point $(h,k)$ defines the ellipses center.
• Points $(\pm a + h,k)$ defines the ellipses domain, and horizontal endpoints - i.e. horizontal stretch.
• Points $(h,\pm b + k)$ defines the ellipses range, and vertical endpoints - i.e. vertical stretch.

#### Circles

General formula:

• $(x-h)^2 + (y-k)^2 = r^2$

General Notes:

• Point $(h,k)$ defines the circles center.
• Points $(\pm r + h,k)$ defines the circles domain - i.e. stretch.
• Points $(h,\pm r + k)$ defines the circles range - i.e. stretch.
• A circle is a subset of an ellipse, such that $a = b = r$.

#### Hyperbolas

General formulae:

• $\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1$
• $\frac{(y-k)^2}{b^2} - \frac{(x-h)^2}{a^2} = 1$

General Notes:

• Point $(h,k)$ defines the hyperbolas center.
• Points $(\pm a + h,k)$ defines the hyperbolas domain, $[\pm a + h, \pm \infty)$.
• The switch in positions of the fractions containing x and y, indicate the type of hyperbola - i.e. vertical or horizontal. The hyperbola is horizontal in the first, and negative in the second of the General hyperbolic formulae above.
• Graphs $y=\pm (\pm a + h,k)$ defines the hyperbolas domain $[\pm a + h, \pm \infty)$.

### Trignometric 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.

#### 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.

#### 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.

#### Arcsin

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

#### Arccos

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

#### Arctan

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