Periodicity of Trigonometric Functions

1. Definition of Periodicity

The period of a function is the smallest positive number T such that the function repeats its values when the input is increased by T.

f(x + T) = f(x)

If such a number exists, the function is called periodic.

2. Period of the Sine Function

The sin x function is one of the most important periodic functions.

sin(x + 2π) = sin x

Therefore, the period of the sine function is:

T = 2π

3. Period of the Cosine Function

The cos x function also repeats its values every 2π.

cos(x + 2π) = cos x

Thus, the period of the cosine function is:

T = 2π

4. Period of the Tangent Function

The tan x function has a shorter period compared to sine and cosine.

tan(x + π) = tan x

Hence, the period of the tangent function is:

T = π

5. Period of the Cotangent Function

The cot x function is also periodic with a period equal to π.

cot(x + π) = cot x

T = π

6. Period of Trigonometric Functions with Coefficients

When a coefficient a is multiplied by the variable inside a trigonometric function, the period changes.

For sine and cosine functions:

sin(ax), cos(ax) → T = 2π / |a|

For tangent and cotangent functions:

tan(ax), cot(ax) → T = π / |a|

7. Importance of Periodicity in Problem Solving

Understanding the period of trigonometric functions is essential for solving trigonometric equations, sketching graphs, and analyzing repetitive behaviors in mathematical and real-world applications.

Conclusion

The periodicity of trigonometric functions determines how often a function repeats its values. Sine and cosine have a period of 2π, while tangent and cotangent have a period of π. Mastering this concept is a key step in advanced trigonometry.