Graphs of trigonometric functions

Basic Trigonometric Functions

Sine (sin), cosine (cos), and tangent (tan) are fundamental trigonometric functions.
These functions represent the ratio of sides in a right-angled triangle.

Periodic Nature of Trigonometric Functions

All trigonometric functions are periodic, meaning they repeat their values in regular intervals or periods.
The sine and cosine functions are periodic with a period of 2π, while the tangent function is periodic with a period of π.

Graphs of Sin and Cos

The graph of sin(x) starts at the origin (0,0), reaches its peak at π/2, comes back to 0 at π, goes to its lowest at 3π/2, and returns to 0 at 2π, repeating the pattern thereafter.
The graph of cos(x) starts from a peak at (0,1), moves to 0 at π/2, reaches its lowest point at π, goes back to zero at 3π/2, reaches a peak at 2π, and continues repeating the pattern.

Graph of Tan

The graph of tan(x) is different from sin(x) and cos(x). It starts from zero, increases indefinitely, approaches vertical asymptotes at ±π/2, becomes undefined at these points, decreases from there, reaches zero at π, and repeats the pattern.
Unlike sin and cos, tan has a period of π instead of 2π.

Transformations of Trigonometric Functions

Amplitude change: The graphs of sin and cos can change amplitude (height) with a multiplier called the amplitude factor. For instance, in f(x) = Asin(x) or f(x) = Acos(x), the amplitude is

Frequency change: A horizontal compression or stretch in the graph by a factor B creates a frequency change, as seen in f(x) = sin(Bx) or f(x) = cos(Bx).
Phase shift or horizontal shift: This is a movement of the entire graph horizontally. It’s introduced by a constant added or subtracted from x, as in f(x) = sin(x + C).
Vertical shift: This is a movement of the entire graph up or down and is introduced by a constant added or subtracted from the entire function, as in f(x) = sin(x) + D.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arcsin or sin^-1, arccos or cos^-1, and arctan or tan^-1 are used to reverse sine, cosine, and tangent.
These functions allow you to input a ratio and get an angle that would give you that ratio.