Trigonometric Graphs

Section 1: Basic Understanding

Trigonometric graphs are visuals that represent the sine, cosine, and tangent functions.
The x-axis represents the angle in radians or degrees, while the y-axis represents the value of the trigonometric function.
The graphs oscillate between a maximum and a minimum, usually between -1 and 1 for both sine and cosine.

Section 2: Sine and Cosine Graphs

The sine graph starts at 0, rises to 1 at 90 degrees (π/4 rad), falls to 0 at 180 degrees (π rad), decreases to -1 at 270 degrees (3π/4 rad) and returns to 0 at 360 degrees (2π rad).
The cosine graph starts at 1, drops to 0 at 90 degrees (π/4 rad), falls to -1 at 180 degrees (π rad), rises to 0 at 270 degrees (¾π rad) and returns to 1 at 360 degrees (2π rad).
Sine and cosine graphs repeat every 360 degrees (2π rad), which makes them periodic functions.

Section 3: Tangent Graph

The tangent graph starts at 0, rises to a value, decreases, and rises again.
Tangent has vertical asymptotes at ±90 degrees and every 180 degrees thereafter.
The tangent graph is also periodic, but it repeats every 180 degrees (π rad).

Section 4: Transformations of Trigonometric Graphs

Trigonometric functions can be transformed by changing their amplitude, period, phase shift, or vertical shift.
Amplitude is the ‘height’ of the graph, and a change in amplitude stretches or squashes the graph vertically.
Period is the ‘length’ of one complete cycle of the graph; altering the period compresses or stretches the graph horizontally.
Phase shift or horizontal shift moves the graph left or right.
Vertical shift moves the graph up or down.

Section 5: Inverse Trigonometric Graphs

Inverse trigonometric functions, such as arcsin (sin^-1), arccos (cos^-1), and arctan (tan^-1), have graphs which are reflections of the original functions in the line y = x.
Inverse trigonometric functions are useful in determining an angle given the sides of a right triangle, and their graphs exhibit the related values of the angles.