Graphs of Trigonometric Functions

Trigonometric graphs are related to the unit circle. As a point moves counterclockwise around the circle, its vertical projection traces the sine wave.

Similarly, the horizontal projection of the point’s motion around the circle produces the cosine wave.

Sine and cosine graphs have a period of 2π, a domain of all real numbers, and a range from -1 to 1.

On the unit circle, the intersection of a vertical tangent line and a secant on the circle that passes through the origin traces the tangent function graph. The tangent function is defined everywhere except where the cosine is zero. Its range is all real numbers.

The cosecant graph repeats every 2π. It is undefined wherever the sine is zero. Its values are always less than -1 or greater than 1.

The secant graph also has a 2π period. It is undefined where cosine is zero and has the same range as cosecant.

The cotangent graph is undefined where the sine is zero and has a range of all real numbers.

Trigonometric graphs model periodic motion—like the height of a passenger on a Ferris wheel rising and falling over time traces a sine wave as the wheel rotates around its center.