When you graph trigonometric functions, you discover they are periodic; that is, they produce results that repeat predictably. To find the period of a given function, you need some familiarity with each one and how variations in their use affect the period. Once you recognize how they work, you can pick apart trig functions and find the period with no trouble.
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The period of the sine and cosine functions is 2π (pi) radians or 360 degrees. For the tangent function, the period is π radians or 180 degrees.
Defined: Function Period
When you plot them on a graph, the trigonometric functions produce regularly-repeating wave shapes. Like any wave, the shapes have recognizable features such as peaks (high points) and troughs (low points). The period tells you the angular “distance” of one full cycle of the wave, usually measured between two adjacent peaks or troughs. For this reason, in math, you measure a function’s period in angle units. For example, starting at an angle of zero, the sine function produces a smooth curve that rises to a maximum of 1 at π / 2 radians (90 degrees), crosses zero at π radians (180 degrees), decreases to a minimum of −1 at 3π / 2 radians (270 degrees) and reaches zero again at 2π radians (360 degrees). After this point, the cycle repeats indefinitely, producing the same features and values as the angle increases in the positive x direction.
Sine and Cosine
The sine and cosine functions both have a period of 2π radians. The cosine function is very similar to the sine, except that it is “ahead” of the sine by π / 2 radians. The sine function takes the value of zero at zero degrees, where as the cosine is 1 at the same point.
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The Tangent Function
You get the tangent function by dividing sine by cosine. Its period is π radians or 180 degrees. The graph of tangent (x) is zero at angle zero, curves upward, reaches 1 at π / 4 radians (45 degrees), then curves upward again where it reaches a divide-by-zero point at π / 2 radians. The function then becomes negative infinity and traces out a mirror image below the y axis, reaching −1 at 3π / 4 radians, and crosses the y axis at π radians. Although it has x values at which it becomes undefined, the tangent function still has a definable period.
Secant, Cosecant and Cotangent
The three other trig functions, cosecant, secant and cotangent, are the reciprocals of sine, cosine and tangent, respectively. In other words, cosecant (x) is 1 / sin(x), secant(x) = 1 / cos(x) and cot(x) = 1 / tan(x). Although their graphs have undefined points, the periods for each of these functions is the same as for sine, cosine and tangent.
Period Multiplier and Other Factors
By multiplying the x in a trigonometric function by a constant, you can shorten or lengthen its period. For example, for the function sin(2_x_), the period is one-half of its normal value, because the argument x is doubled. It reaches its first maximum at π / 4 radians instead of π / 2, and completes a full cycle in π radians. Other factors you commonly see with trig functions include changes to the phase and amplitude, where the phase describes a change to the starting point on the graph, and amplitude is the function's maximum or minimum value, ignoring the negative sign on the minimum. The expression, 4 × sin(2_x_ + π), for example, reaches 4 at its maximum, due to the 4 multiplier, and starts by curving downward instead of upward because of the π constant added to the period. Note that neither the 4 nor the π constants affect the function’s period, only its starting point and maximum and minimum values.