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Chapter 1: Problem 4

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

Short Answer

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

Step by step solution

01

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.
02

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 2蟺. This means that their values repeat after every 2蟺 interval. In other words, sin(x + 2蟺) = sin(x) and cos(x + 2蟺) = cos(x) for all values of x.
03

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 蟺, which means that its values repeat every 蟺 interval. In other words, tan(x + 蟺) = 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 蟺. So, we have cot(x + 蟺) = cot(x) for all values of x where cot(x) is defined.
04

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 2蟺. This means that its values repeat after every 2蟺 interval. In other words, sec(x + 2蟺) = 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 2蟺. So, we have csc(x + 2蟺) = 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): 2蟺 - tan(x) and cot(x): 蟺 - sec(x) and csc(x): 2蟺

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