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Solving Trigonometric Equations Using Double Angle Identity: A Comprehensive Guide

January 07, 2025Science2605
Solving Tr

Solving Trigonometric Equations Using Double Angle Identity: A Comprehensive Guide

In solving complex trigonometric equations, the application of the double angle identity can be particularly beneficial. One common identity is cos 2x 2cos2x - 1. This identity allows for the reduction of terms involving higher powers of cosine, making the equation easier to solve. Let's explore how we can use this identity in various scenarios.

Using the Double Angle Identity to Solve Equations

Given an equation, we can use the double angle identity to simplify it and solve for the variable in question. For example, consider the equation:

cos 4θ 2cos22θ - 1 .

We can apply the double angle identity step-by-step to simplify the equation.

Step 1: Apply the Double Angle Identity

First, let's start by expressing cos 4θ in terms of cos 2θ using the double angle identity:

cos 4θ 2cos22θ - 1

Loading this into our original equation, we have:

2cos22θ - 1 2cos22θ - 1

Step 2: Simplify Further

By now, we recognize that both sides of the equation are identical, indicating that this step did not introduce any simplification. However, we can further simplify by taking the square root of both sides. Since square roots can result in positive and negative values, we must consider both possibilities:

sqrt{2cos22θ - 1} sqrt{2cos22θ - 1}

At this point, we can simplify it using the formula for the square root of a squared term:

sqrt{2cos22θ} sqrt{2cos22θ - 1 1}

Step 3: Using the Given Values

To validate our solution, let's assume θ 0. This simplification is often used when time is limited and we want a quick solution:

sqrt{2cos20} sqrt{2cos20} 2 2 cosθ

By taking the square root on both sides, we derive:

2cosθ 2

Dividing both sides by 2, we get:

cosθ 1

Therefore, the correct answer is option C, which is:

2cosθ

Additional Examples and Insights

Example 1: Simplifying Using Different Values

Let us take the values of θ as 30deg;, 60deg;, and 90deg; to ensure the solution is consistent across different angles:

For θ 30o: cos 30o sqrt(3)/2, 2cos30o sqrt(3)
For θ 60o: cos 60o 1/2, 2cos60o 1
For θ 90o: cos 90o 0, 2cos90o 0

In all cases, the application of the double angle identity and the final value of 2cosθ holds.

Example 2: Double Angle Formula for Cosine

We can use the double angle formula for cosine, which is:

cos 2x 2cos2x - 1

By applying this to different angles, we can verify the consistency of our solution:

For x 2θ, we have:
cos 4θ 2cos22θ - 1
cos 4θ 4cos22θ - 1 (using the identity twice)

Once again, taking the square root of both sides:

sqrt{4cos22θ - 1} sqrt{2cos22θ - 1}

This confirms that our solution is consistent with the double angle identity.

Conclusion

By applying the double angle identity cos 2x 2cos2x - 1, we can simplify and solve complex trigonometric equations efficiently. The key steps involve substituting the identity, simplifying, and verifying the solution with different angles. Understanding these steps can significantly enhance one's problem-solving skills in trigonometry.