Circular Functions and Trigonometry

IB Higher Level Mathematics : Core

3. Circular Functions and Trigonometry

3.1

The circle : radian measure of angles; length of an arc; area of a sector.

3.2

Definition of ( cos q, sin q ) in terms of the unit circle.
The Pythagorean identities:

cos²q + sin²q = 1

1 + tan²q = sec ²q

1 + cot²q = cosec²q

3.3

The six circular functions :
x

sin x, x

cos x, x

tan x,
x

csc x, x

sec x, x

cot x;
their domains and ranges; their periodic nature, and their graphs.
The inverse functions
x

arcsin x, x

arccos x, x

arctan x;
their domains and ranges, and their graphs.

See The Trigonometric Functions and Calculus for Liberal Arts and Business Majors,
a text resource written by Stefan Waner and Steven Constable of Hofstra University.

At y = sin x, a JAVA applet traces out the sine curve as a point moves around the unit circle. The cosine function is treated at y = cos x. The process is carried further with y = sin ax,
y = a sin b(x - c), and y = sin x + cos x. Also see Six Trig Functions. These applets are part of the Manipula Math with Java site, from Japan.

A different animation of the sine function, together with its gradient and derivative, can be seen at Calculus Animations.

The sine and cosine functions.
See The Trigonometric Functions, one of the sections from Trigonometric Functions and Calculus, from Stefan Waner and Steven Costenoble of Hofstra University. Used with permission.

3.4

Addition, double-angle and half-angle formulae:
sin (A+B), etc.; sin 2A, etc.; sin ½ A, etc.;

The compound formula
acosx ± bsinx = Rcos (x

Simple trigonometric equations are dealt with at Trigonometric Equations, from the University of Texas El Paso's S.O.S. MATHematics site.

3.5

Composite functions of the form
f(x) = asinb (x + c) ; solutions of f(x) = k in a given finite interval.

Solution of equations leading to quadratic or linear equations in sin x, etc.
Graphical interpretation of the above.

3.6

Solution of triangles, using the Cosine Rule, the Sine Rule, and the area of a triangle as ½ ab sinC