Solve 6 Cos ⁡ ( 2 X ) = 6 Sin ⁡ 2 ( X ) + 4 6 \cos(2x) = 6 \sin^2(x) + 4 6 Cos ( 2 X ) = 6 Sin 2 ( X ) + 4 For All Solutions 0 ≤ X \textless 2 Π 0 \leq X \ \textless \ 2\pi 0 ≤ X \textless 2 Π . X = □ X = \, \square X = □ Give Your Answers Accurate To At Least 2 Decimal Places, As A List Separated By Commas.

Mar 11, 2025 by ADMIN 312 views

**Solving Trigonometric Equations: A Step-by-Step Guide**

Introduction

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we will focus on solving the equation $6 \cos(2x) = 6 \sin^2(x) + 4$ for all solutions $0 \leq x \ \textless \ 2\pi$ . We will use various trigonometric identities and techniques to simplify the equation and find the solutions.

Step 1: Simplify the Equation

The given equation is $6 \cos(2x) = 6 \sin^2(x) + 4$ . We can start by simplifying the left-hand side of the equation using the double-angle identity for cosine:

\cos(2x) = 1 - 2\sin^2(x)

Substituting this into the original equation, we get:

6(1 - 2\sin^2(x)) = 6 \sin^2(x) + 4

Expanding and simplifying, we get:

6 - 12\sin^2(x) = 6 \sin^2(x) + 4

Step 2: Rearrange the Equation

We can rearrange the equation to isolate the sine term:

-12\sin^2(x) - 6 \sin^2(x) = 4 - 6

Combine like terms:

-18\sin^2(x) = -2

Divide both sides by -18:

\sin^2(x) = \frac{1}{9}

Step 3: Solve for Sine

We can take the square root of both sides to solve for sine:

\sin(x) = \pm \sqrt{\frac{1}{9}}

Simplifying, we get:

\sin(x) = \pm \frac{1}{3}

Step 4: Find the Solutions

We can use the inverse sine function to find the solutions:

x = \sin^{-1}\left(\pm \frac{1}{3}\right)

Using a calculator, we can find the solutions:

x \approx 0.3398, 1.8239, 3.1416, 4.4577

Conclusion

In this article, we solved the equation $6 \cos(2x) = 6 \sin^2(x) + 4$ for all solutions $0 \leq x \ \textless \ 2\pi$ . We used various trigonometric identities and techniques to simplify the equation and find the solutions. The solutions are:

x \approx 0.3398, 1.8239, 3.1416, 4.4577

Discussion

Trigonometric equations are a fundamental concept in mathematics, and solving them requires a deep understanding of trigonometric functions and their properties. In this article, we used various trigonometric identities and techniques to simplify the equation and find the solutions. The solutions we found are accurate to at least 2 decimal places.

Additional Resources

For more information on trigonometric equations and their solutions, please refer to the following resources:

Final Answer

The final answer is:

x \approx 0.3398, 1.8239, 3.1416, 4.4577$<br/> **Solving Trigonometric Equations: A Q&A Guide** =====================================================

Introduction

In our previous article, we solved the equation $6 \cos(2x) = 6 \sin^2(x) + 4$ for all solutions $0 \leq x \ \textless \ 2\pi$ . We used various trigonometric identities and techniques to simplify the equation and find the solutions. In this article, we will answer some frequently asked questions about solving trigonometric equations.

Q: What are some common trigonometric identities used to solve equations?

A: Some common trigonometric identities used to solve equations include:

Double-angle identity for sine: $\sin(2x) = 2\sin(x)\cos(x)$
Double-angle identity for cosine: $\cos(2x) = 1 - 2\sin^2(x)$
Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$
Sum and difference identities: $\sin(x+y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ and $\sin(x-y) = \sin(x)\cos(y) - \cos(x)\sin(y)$

Q: How do I simplify a trigonometric equation?

A: To simplify a trigonometric equation, you can use various techniques such as:

Factoring: Factor out common terms or expressions
Combining like terms: Combine terms with the same variable or expression
Using trigonometric identities: Use trigonometric identities to simplify expressions
Using algebraic manipulations: Use algebraic manipulations such as adding or subtracting terms to simplify the equation

Q: How do I solve a trigonometric equation?

A: To solve a trigonometric equation, you can use various techniques such as:

Using inverse trigonometric functions: Use inverse trigonometric functions such as $\sin^{-1}(x)$ or $\cos^{-1}(x)$ to find the solutions
Using trigonometric identities: Use trigonometric identities to simplify the equation and find the solutions
Using algebraic manipulations: Use algebraic manipulations such as adding or subtracting terms to simplify the equation and find the solutions

Q: What are some common mistakes to avoid when solving trigonometric equations?

A: Some common mistakes to avoid when solving trigonometric equations include:

Not using the correct trigonometric identity: Using the wrong trigonometric identity can lead to incorrect solutions
Not simplifying the equation enough: Failing to simplify the equation enough can lead to incorrect solutions
Not checking the solutions: Failing to check the solutions can lead to incorrect solutions

Q: How do I check the solutions of a trigonometric equation?

A: To check the solutions of a trigonometric equation, you can use various techniques such as:

Plugging in the solutions: Plug in the solutions into the original equation to check if they are true
Using a calculator: Use a calculator to check if the solutions are true
Graphing the equation: Graph the equation to check if the solutions are true