Graphs of Trigonometric Ratios (Periodicity)

Trigonometric functions are fundamental in mathematics, as they relate the angles of a triangle to the lengths of its sides. These functions are periodic, meaning they repeat their values in regular intervals. Understanding the graphs of trigonometric ratios is crucial for solving problems in various fields such as physics, engineering, and computer science.

Basic Trigonometric Functions

The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot). Here, we will focus on the periodicity of the sine, cosine, and tangent functions, as they are the primary functions from which the others are derived.

Sine and Cosine Functions

The sine and cosine functions have a period of (2\pi) radians (or 360 degrees), which means they repeat their values every (2\pi) radians.

Sine Function (sin)

The sine function, (y = \sin(x)), starts at 0, reaches a maximum of 1 at (\pi/2) radians, returns to 0 at (\pi) radians, reaches a minimum of -1 at (3\pi/2) radians, and then returns to 0 at (2\pi) radians.

Cosine Function (cos)

The cosine function, (y = \cos(x)), starts at 1, decreases to 0 at (\pi/2) radians, reaches a minimum of -1 at (\pi) radians, returns to 0 at (3\pi/2) radians, and then goes back to 1 at (2\pi) radians.

Tangent Function

The tangent function, (y = \tan(x)), has a period of (\pi) radians (or 180 degrees). Unlike sine and cosine, the tangent function does not have a maximum or minimum value, as it approaches infinity when the cosine of the angle is zero.

Graphs and Periodicity

The periodic nature of these functions is best understood by looking at their graphs.

Graph of Sine Function

y = sin(x)

Graph of Cosine Function

y = cos(x)

Graph of Tangent Function

y = tan(x)

Important Points and Differences

Function	Period	Starts at	Key Points
sin(x)	(2\pi)	0	(y = 0) at (x = 0, \pi, 2\pi); (y = 1) at (x = \frac{\pi}{2}); (y = -1) at (x = \frac{3\pi}{2})
cos(x)	(2\pi)	1	(y = 1) at (x = 0, 2\pi); (y = 0) at (x = \frac{\pi}{2}, \frac{3\pi}{2}); (y = -1) at (x = \pi)
tan(x)	(\pi)	Undefined	Discontinuities at (x = \frac{\pi}{2} + k\pi), where (k) is an integer

Examples

Example 1: Sine Function

Consider the function (y = \sin(x)). At (x = \frac{\pi}{2}), (y = 1). After a period of (2\pi), at (x = \frac{\pi}{2} + 2\pi), (y) will again be 1.

Example 2: Cosine Function

For the function (y = \cos(x)), at (x = \pi), (y = -1). After a period of (2\pi), at (x = \pi + 2\pi), (y) will again be -1.

Example 3: Tangent Function

With the function (y = \tan(x)), at (x = \frac{\pi}{4}), (y = 1). After a period of (\pi), at (x = \frac{\pi}{4} + \pi), (y) will again be 1.

Conclusion

Understanding the periodicity of trigonometric functions is essential for interpreting their graphs and applying them to real-world problems. The sine and cosine functions repeat every (2\pi) radians, while the tangent function repeats every (\pi) radians. These patterns are the foundation for analyzing oscillatory motion, waves, and many other phenomena in science and engineering.