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

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Circular Functions
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Coordinate Geometry Complex Numbers


Preface[edit]

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[edit]

Sin[edit]

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[edit]

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[edit]

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[edit]

Also known as Sin^-1 or sin^-

Arccos[edit]

Also known as Cos^-1 or cos^-

Arctan[edit]

Also known as Tan^-1 or tan^-

Examples[edit]

Graphing Functions[edit]

General Method[edit]

Trigonometric Functions Method[edit]
Reciprocal Trigonometric Functions Method[edit]
Inverse Trigonometric Functions Method[edit]

Sin[edit]

Cos[edit]

Tan[edit]

Arcsin[edit]

Arccos[edit]

Arctan[edit]