Solving the Equation y sin cosx: A Comprehensive Guide

In this article, we explore the solution to the trigonometric equation y sin cosx. This equation involves the composition of two trigonometric functions: sine and cosine, which are periodic and bounded. Understanding these functions and their properties is crucial for solving such equations. We will walk through a step-by-step approach to find x for a given value of y, along with examples and insights on periodicity and complementary angles.

Understanding the Function

The function y sin cosx is a composite function formed by the sine and cosine functions. The cosine function, cosx, always produces values within the range [-1, 1]. Similarly, the sine function, sint, also produces values within a similar range [-1, 1] when t is any real number.

Range of y

Given the properties of cosine, the values of sincosx will also be constrained to values between -sin(1) and sin(1). This is because the sine function takes values within [-1, 1], and when applied to a value in [-1, 1], the range remains [-sin(1), sin(1)]. Therefore, the range of y is approximately:

sin(-1) ≈ -0.8415 sin(1) ≈ 0.8415

Finding x for a Given y

To find the values of x for a given y, follow these steps:

Set the Equation: Start with y sin cosx. Solve for cosx: Express cosx in terms of y using the inverse sine function: cosx sin-1y or x ± cos-1(sin-1y). Ensure that y is within the range [-0.8415, 0.8415] to find valid solutions. Calculate x: Use the inverse cosine function to solve for x while accounting for the periodic nature of the cosine function. The general solution involves accounting for integer multiples of 2π: x ± cos-1(sin-1y) 2nπ.

Example: If you want to solve for y 0.5:

Find k: k sin-1(0.5) π/6. Now solve for x: x ± cos-1(π/6) 2nπ.

Conclusion: Solving y sin cosx involves ensuring y is within the range [-0.8415, 0.8415], finding cosx using the inverse sine function, and solving for x using the inverse cosine function, accounting for periodic solutions.

Complementary Angles and Periodicity

Another approach to solving sin y cos x is to recognize that the sine and cosine functions are complementary. From the identity sin y cos(π/2 - y), we can derive:

sin y cos x, which implies y π/2 - x 2nπ or π - y π/2 x 2nπ

Sketching the graph of these functions can help visualize and confirm the solutions. This periodicity allows us to find multiple solutions for x and y for any given value within their respective ranges.