Trigonometry: Periodic Functions (Class 11th Maths)

*Periodic Functions* in trigonometry are functions that repeat their values after a fixed interval. This interval is called the *period* of the function. In simpler terms, if you graph a periodic function, it will create a pattern that repeats itself over and over.

*Key Trigonometric Functions and Their Periods:*

*Sine (sin x):* Period = 2π
*Cosine (cos x):* Period = 2π
*Tangent (tan x):* Period = π
*Cosecant (csc x):* Period = 2π
*Secant (sec x):* Period = 2π
*Cotangent (cot x):* Period = π

*Understanding Period:*
*Sine and Cosine:* These functions oscillate between 1 and -1, repeating their pattern every 2π units on the x-axis.
*Tangent and Cotangent:* These functions have a vertical asymptote at x = π/2 + nπ (where n is an integer). They repeat their pattern every π units.
*Cosecant and Secant:* These functions are the reciprocals of sine and cosine, respectively, and have the same periods as their corresponding functions.

*Properties of Periodic Functions:*

*Symmetry:* Many periodic functions exhibit symmetry. For example, sine and cosine are even functions (symmetric about the y-axis), while tangent and cotangent are odd functions (symmetric about the origin).
*Amplitude:* The amplitude of a periodic function is half the distance between its maximum and minimum values.
*Phase Shift:* The phase shift is the horizontal displacement of a periodic function from its standard position.

*Applications of Periodic Functions:*

*Modeling Periodic Phenomena:* Trigonometric functions are used to model various periodic phenomena in nature and engineering, such as sound waves, alternating currents, and the motion of a pendulum.
*Solving Trigonometric Equations:* Periodic functions are involved in solving trigonometric equations, which have applications in fields like physics, engineering, and navigation.

By understanding the properties and applications of periodic functions, you can effectively use trigonometry to solve problems and model real-world phenomena.

Trigonometry#Periodic Functions#Class 11th Maths#Sine#Cosine#Tangent#Cosecant#Secant#Cotangent#Period#Amplitude#Phase Shift#Symmetry#Trigonometric Equations#Modeling Periodic Phenomena#Sound Waves#Alternating Currents#Pendulum Motion.