Graph of ycosx: A Comprehensive Guide

Introduction

The cosine function, denoted as ycosx, is one of the basic trigonometric functions. This article aims to guide you through understanding and visualizing the graph of ycosx by breaking it down into manageable steps, explaining the process in detail.

Understanding the Cosine Function

The cosine function is a periodic function with a period of 2π. It represents the x-coordinate of a point on the unit circle, which is the circle centered at the origin with a radius of one.

Graphing ycosx: Step-by-Step Approach

Case 1: y0

Consider the equation ycosx. The first step is to determine the points where y0. This occurs where the cosine value is zero. From trigonometry, we know that cosx0 at xkπ/2, where k is an odd integer. For example, xπ/2 and x3π/2.

The interval where cosx is non-negative is [2nπ-π/2, 2nπ π/2] for any integer n. We can express the desired points as:

For yge;0, x is in the interval [2nπ-π/2, 2nπ π/2].

To verify, test a few values of x within this interval, e.g., xπ, x3π/2. For these values, cosx 0.

Case 2: cosx is Non-negative

When cosx is non-negative, the graph of ycosx lies in the intervals [2nπ-π/2, 2nπ π/2]. This can be clearly visualized by plotting points within these intervals.

For example, when xπ, cos(π) -1, which is negative. This means that for x in the interval (2nπ-π/2, 2nπ π/2), cosx is not included at the endpoints.

Case 3: cosx is Non-positive

For y 0, we analyze y-cosx. The cosine function is symmetric, meaning cosx is non-positive in the intervals [2nπ π/2, 2nπ 3π/2].

In these intervals, the value of cosx ranges from 0 to -1. Therefore, y-cosx ranges from 0 to 1. The specific intervals for x are [2nπ π/2, 2nπ 3π/2].

To verify, test a few values of x in the interval, e.g., x3π/2, x5π/2. For these values, -cos(3π/2) 1 and -cos(5π/2) 1, which are non-positive.

Conclusion

In summary, the graph of ycosx is a repeating wave pattern with maxima at 1 and minima at -1. The function is symmetric and periodic with a period of 2π. Understanding the intervals where cosx is non-negative and non-positive helps in visualizing the complete graph of ycosx.

By following this step-by-step approach, you can more easily grasp and visualize the graph of the cosine function.