Graphs of Trigonometric Functions

Function of an angle.

Trigonometric functions are fundamental to understanding many phenomena in the natural and physical sciences. They are used to model periodic phenomena such as sound and light waves, describe the motion of pendulums, and much more. This unit will delve into the graphs of these functions and how they can be transformed.

Understanding the Graphs of Sine, Cosine, and Tangent

The sine, cosine, and tangent functions are the three primary trigonometric functions. They are defined using the ratios of the sides of a right triangle, but they can also be defined using the unit circle, which provides a more comprehensive view of these functions and their graphs.

• The sine function, denoted by sin(x), produces a wave-like graph that oscillates between -1 and 1. The graph is periodic with a period of 2π.

• The cosine function, denoted by cos(x), also produces a wave-like graph that oscillates between -1 and 1. It is identical to the sine function but shifted π/2 units to the left. It also has a period of 2π.

• The tangent function, denoted by tan(x), produces a graph with vertical asymptotes at odd multiples of π/2. It has a period of π.

Transformations of Trigonometric Graphs

Just like other functions, trigonometric functions can undergo transformations such as shifts, stretches, and reflections. These transformations can change the amplitude, period, and phase of the function.

• Vertical and horizontal shifts move the graph up, down, left, or right. A vertical shift changes the midline of the function, while a horizontal shift changes the phase of the function.

• Reflections flip the graph over the x-axis or y-axis. A reflection over the x-axis changes the sign of the function, while a reflection over the y-axis changes the direction of the function.

• Stretches change the amplitude or period of the function. A vertical stretch changes the amplitude, while a horizontal stretch changes the period.

Graphing Inverse Trigonometric Functions

Inverse trigonometric functions are the inverses of the sine, cosine, and tangent functions. They are denoted by arcsin(x), arccos(x), and arctan(x) or sin^(-1)(x), cos^(-1)(x), and tan^(-1)(x).

• The arcsine function, denoted by arcsin(x) or sin^(-1)(x), produces a graph that increases from -π/2 to π/2.

• The arccosine function, denoted by arccos(x) or cos^(-1)(x), produces a graph that decreases from 0 to π.

• The arctangent function, denoted by arctan(x) or tan^(-1)(x), produces a graph that increases from -π/2 to π/2, but unlike the arcsine function, it has horizontal asymptotes at these values.

By understanding the graphs of trigonometric functions and their transformations, we can better understand and model the world around us.