How to Find the Stationary Points of y sin(x) - 2 Using Calculus: A Detailed Guide

When solving problems involving trigonometric functions to find stationary points analytically, one often encounters expressions like y sin(x) - 2. This article delves into the intricate steps involved in determining the stationary points of such a function and is designed to cater to students, educators, and professionals in the field of mathematics and related disciplines.

Understand the Basics of Stationary Points and Trigonmetric Functions

In the context of calculus, a stationary point refers to a point on the graph of a function where the derivative of the function is zero or undefined. Trigonometric functions, like the sine function, sin(x), play a significant role in many mathematical applications, from physics to engineering. This article focuses on the method of finding stationary points for the y f(x) sin(x) - 2 function.

Deriving the Function

To find the stationary points of y sin(x) - 2, we start by differentiating the function with respect to x. This is a fundamental step in optimization problems and critical for understanding where the function might have local maxima, minima, or points of inflection.

The derivative of y sin(x) - 2 with respect to x is given by:

dy/dx cos(x)

This derivative equals zero at certain points on the graph, known as stationary points.

Setting the Derivative to Zero

To find the stationary points, we set cos(x) equal to zero:

cos(x) 0

The equation cos(x) 0 has solutions at specific values of x that can be determined using the unit circle and properties of the cosine function:

x π/2, 3π/2, 5π/2, ...

These points represent locations where the function y sin(x) - 2 has horizontal tangents (stationary points).

Evaluating the Function at the Stationary Points

Now that we know the x-values where the stationary points occur, we need to evaluate the original function y sin(x) - 2 at these points to find the corresponding y-values:

For x π/2:

y sin(π/2) - 2 1 - 2 -1

For x 3π/2:

y sin(3π/2) - 2 -1 - 2 -3

And so on for other values of x. However, the point at x π/2, y -1 is often of primary interest for optimization and graphical analysis.

Conclusion: Understanding the Graph and Applications

Understanding how to find and interpret stationary points is crucial in many fields. For the function y sin(x) - 2, the point (π/2, -1) is a critical point in the graph, indicating a local minimum or maximum in the context of optimization problems.

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