Question

Explain what is meant by the period of a trigonometric function. What are the periods of the six trigonometric functions?

Short answer

Answer: The periods of the six trigonometric functions are as follows: - sin(x) and cos(x): 2π - tan(x) and cot(x): π - sec(x) and csc(x): 2π

Step-by-step solution

Definition of Period

The period of a trigonometric function is the smallest positive value P such that the function repeats itself after every interval of length P. In other words, a function f(x) has period P if f(x+P) = f(x) for all values of x in the domain of the function.

Sin(x) and Cos(x) Periods

The two most common trigonometric functions are sine and cosine. The sine function, sin(x), and the cosine function, cos(x), both have a period of 2π. This means that their values repeat after every 2π interval. In other words, sin(x + 2π) = sin(x) and cos(x + 2π) = cos(x) for all values of x.

Tan(x) and Cot(x) Periods

The tangent function, tan(x), is defined as the ratio of the sine and cosine functions: tan(x) = sin(x)/cos(x). The tangent function has a period of π, which means that its values repeat every π interval. In other words, tan(x + π) = tan(x) for all values of x where tan(x) is defined. The cotangent function, cot(x), is the reciprocal of the tangent function: cot(x) = 1/tan(x) or cot(x) = cos(x)/sin(x). The cotangent function also has a period of π. So, we have cot(x + π) = cot(x) for all values of x where cot(x) is defined.

Sec(x) and Csc(x) Periods

The secant function, sec(x), is the reciprocal of the cosine function: sec(x) = 1/cos(x). Like the cosine function, the secant function also has a period of 2π. This means that its values repeat after every 2π interval. In other words, sec(x + 2π) = sec(x) for all values of x where sec(x) is defined. The cosecant function, csc(x), is the reciprocal of the sine function: csc(x) = 1/sin(x). Like the sine function, the cosecant function also has a period of 2π. So, we have csc(x + 2π) = csc(x) for all values of x where csc(x) is defined. In summary, the periods of the six trigonometric functions are: - sin(x) and cos(x): 2π - tan(x) and cot(x): π - sec(x) and csc(x): 2π