Solving Trigonometric Equations: A Comprehensive Guide

Trigonometric equations can often be quite challenging due to the multiple steps and identities involved. However, by leveraging trigonometric identities and understanding the unit circle, solving these equations becomes much more manageable. In this guide, we will discuss a detailed method to solve the equation sin 2x - cos x 0.

Understanding Trigonometric Identities

The main idea behind solving such equations is to use the trigonometric identities. In particular, remember the identity sin 2x 2 sin x cos x. This identity will be crucial in breaking down the original equation. Let's start by rewriting the given equation:

#8711;#8711;sin 2x - cos x 0

Step 1: Rewrite the Original Equation

First, we rewrite the original equation using the double-angle identity:

sin 2x - cos x 0

Using sin 2x 2 sin x cos x, we get:

2 sin x cos x - cos x 0

Step 2: Factor the Equation

Next, we factor the equation to simplify the problem:

(2 sin x - 1) cos x 0

This gives us two separate equations to solve:

cos x 0 2 sin x - 1 0, or equivalently, sin x 1/2

Solving Each Equation Separately

Equation 1: cos x 0

Considering the unit circle, we know that:
cos(π/2) 0 and cos(-π/2) 0.
Extending this to all real numbers, the solution set is:

{π/2 kπ | k ∈ Z}

This represents an infinite set of solutions, as the cosine function is periodic with a period of 2π.

Equation 2: sin x 1/2

Using the unit circle again, we find that:
sin(π/6) 1/2 and sin(5π/6) 1/2.
Therefore, the solution set is the union of two sets:

{π/6 2kπ | k ∈ Z} and {5π/6 2kπ | k ∈ Z}

Combining the Solutions

By combining both sets of solutions, we get:

x π/6 2kπ or x 5π/6 2kπ or x π/2 kπ

Important Considerations

It's crucial to remember to include all possible solutions that arise from these equations. Specifically:

sin x sin (180° - x) cos x cos (360° - x)

These identities ensure that we do not miss any potential solutions. Additionally, it's important to include congruences for the solutions, particularly for the periodic nature of the trigonometric functions.

Final Solution Set

The final solution set, considering all possible congruences, is:

x {π/6 2kπ, 5π/6 2kπ, π/2 kπ | k ∈ Z}

Conclusion

By carefully applying trigonometric identities and understanding the unit circle, we can systematically solve trigonometric equations. This guide provides a step-by-step approach to solving sin 2x - cos x 0, demonstrating the use of these fundamental concepts.

Key Takeaways:

Utilize trigonometric identities to simplify the equation. Consider the unit circle and periodicity of trigonometric functions. Include all possible solutions and their congruences.

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