Solving the Trigonometric Equation cos x sin x 1: A Comprehensive Guide

Solving the Trigonometric Equation cos x sin x 1: A Comprehensive Guide

The equation cos x sin x 1 presents a challenge in trigonometry. This article delves into various methods to solve this equation, providing a comprehensive guide for learners and professionals alike. From direct trigonometric properties to algebraic manipulations, we will cover all the necessary steps to find the solutions.

Direct Trigonometric Properties and Solutions

First, we consider the direct properties of trigonometric functions on the unit circle. At x 90° (or π/2), we have:

cos 90° sin 90° 0 * 1 1 This solution is evident because the sine of 90° is 1 and the cosine of 90° is 0. Adding the periodicity of the trigonometric functions, we get: x π/2 2πn, where n is any integer.

However, there are other solutions. Notably, x 0 and x 2πn also satisfy the equation since:

cos 0 * sin 0 1 This means that the product of the cosine and sine of 0 is 0, but when we consider the periodicity, we also have: cos x * sin x 1 cos^2 x - sin^2 x.

Algebraic Manipulation and Solutions

Consider the equation cos x * sin x 1 cos^2 x - sin^2 x. We can use the identity:

cos x / sqrt(1 - cos^2 x) 1 Let y cos x: y / sqrt(1 - y^2) 1 Squaring both sides: y^2 - 2y 1 1 - y^2 Rearranging: 2y^2 - 2y 0 y(y - 1) 0 Thus: y 0 or y 1 cos x 0 or cos x 1 x arccos 0 or x arccos 1 x π/2 2πn or x 0 2πn x π/2 2πn or x 2πn

The basic solution set is x 0 2πn and x π/2 2πn.

General Solutions and Extensions

Given the periodicity of trigonometric functions, we consider:

The solution in the form π/2 2πn is one solution, but there exist infinitely many solutions due to the periodicity. The general solution set S is: S {π/2 2πn : n ∈ Z}. Note 1: The solution in the form 2πn is another obvious particular solution. The overall largest solution set is: S_ext {π/2 2πn : n ∈ Z} ∪ {2πn : n ∈ Z}. Note 3: The solutions 2πn can be extended to kπ in the context of sin x because sin kπ 0 for all integers k, but cos kπ -1 if k is odd. Note 4: The largest possible solution set as per the derived solutions is S_ext {π/2 2πn : n ∈ Z} ∪ {2πn : n ∈ Z}.

Final notes: This solution approach can be extended further by exploring alternative methods such as substituting trigonometric identities or using rational functions. For more detailed approaches, readers are suggested to explore the identities and substitutions described.

Conclusion

Solving the equation cos x sin x 1 involves a combination of trigonometric properties and algebraic manipulations. By understanding the periodicity and using these properties, we can derive the complete solution set. This problem is a great exercise in both trigonometry and algebraic manipulation.