Trigonometric functions exhibit periodic and symmetrical behavior, deeply rooted in the unit circle. The sine and cosine functions correspond to the vertical and horizontal projections, respectively, of a point rotating counterclockwise around the circle. These functions trace smooth, repeating waveforms with identical periods and bounded ranges. The tangent function is defined as the ratio of sine to cosine and produces an unbounded curve that repeats every units, with vertical asymptotes where cosine is zero. Conversely, the cotangent function is the ratio of cosine to sine and behaves similarly but with asymptotes where sine is zero.
The secant and cosecant functions, which are the reciprocals of cosine and sine, respectively, form curves that arch away from the x-axis, becoming undefined at the zeros of their base functions. These graphs display vertical asymptotes and do not intersect the x-axis. Despite their different appearances, all these functions exhibit regular periodicity and distinct symmetries—sine, tangent, and cosecant being odd functions symmetric about the origin, while cosine and secant are even functions symmetric about the y-axis. Their domain restrictions arise from division by zero in their definitions, and their ranges reflect whether they are bounded or unbounded.
Graph Properties of Trigonometric Functions
| Function | Period | Domain | Range |
| Sine | 2 | All real numbers | [−1,1] |
| Cosine | 2 | All real numbers | [−1,1] |
| Tangent | All real numbers x=π2+nπ, where n is any integer | All real numbers | |
| Cotangent | 2 | All real numbers x=nπ, where n is any integer | All real numbers |
| Secant | 2 | All real numbers x=π2+nπ, where n is any integer | All real numbers ≤ –1 or ≤ –1 |
| Cosecant | 2 | All real numbers x=nπ, where n is any integer | All real numbers ≤ –1 or ≤ –1 |
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.
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