## Trigonometric Functions

Graphing Sine Using a Unit Circle Let us start to rotate the point P clockwise over the circle. The y-coordinate of this point varies harmonically with the angle, i.e. .

The diagram shows how to graph the sine function by projecting the y-coordinate of the point P horizontally. When P returns back to its initial position after making a circle the sine function returns back to its initial value of zero.
The Sine Function
Varying the Amplitude Multiplying sin x by a positive constant A causes a vertical expantion or compression of the graph of

y = sin x.

For 0 < A < 1 there is a compression.
For A > 1 there is an expansion.

The Cosine Function
Varying the Amplitude Multiplying cos x by a positive constant A causes a vertical expantion or compression of the graph of

y = cos x.

For 0 < A < 1 there is a compression.
For A > 1 there is an expansion.

Vertical Shift of Sine and Cosine Functions Adding a constant B to sin x or cos x in the equations of the functions
y = sin x + B or y = cos x + B
causes a vertical translation of the graph.

For a positive B > 0 there is a shift upward.
For a negative B < 0 there is a shift downward.

The Tangent Function  Return Last Updated: Jan 18, 1997 Sergey Kiselev, kiselev@msc.cornell.edu