HSC Extension 1 and 2 Mathematics/Trigonometric functions

From Wikibooks, open books for an open world
Jump to: navigation, search

Radian measure of an angle[edit]

2π radians in a revolution

Arc length and area of a sector of a circle[edit]

l = r \theta \;

A = \frac{1}{2}r^2 \theta

  • Where θ is in radians

Area of a segment of a circle[edit]

Minor segment[edit]

A = \frac{1}{2}r^2 ( \theta - \sin \theta )

  • Where θ is in radians

Major segment[edit]

A = \pi r^2 - \frac{1}{2}r^2 ( \theta - \sin \theta )

  • Where θ is in radians

Definitions of trigonometric functions[edit]

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. Trigonometric functions are important in the study of triangles and modeling periodic phenomena, among many other applications.

Symmetry properties of trigonometric functions[edit]

Some exact values[edit]

Graphs of trigonometric functions[edit]

Graphs of y = a sin bx and y = a cos bx[edit]

Graphs of y = a sin b(x + c) and y = a cos b(x + c)[edit]

Graphical solution of equations[edit]

Derivative of sin x and cos x[edit]

\sin 'x = \cos x \;

\cos 'x = - \sin x \;

Derivative of tan x[edit]

\tan 'x = \sec^2 x \;

Derivative of sin (ax + b)[edit]

\sin '(ax + b) = a \cos (ax + b) \;

Derivative of cos (ax + b)[edit]

\cos '(ax + b) = -a \sin (ax + b) \;

Functions defined by integrals (indefinite integrals)[edit]

Primitives of trigonometric functions[edit]

Approximate integration[edit]