Solve The Equation:${ \frac{1+2 \sin X \cos X}{\sin X+\cos X} = \sin X+\cos X }$

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Introduction

In this article, we will delve into the world of trigonometry and explore a fascinating equation involving sine and cosine functions. The equation in question is:

1+2sin⁡xcos⁡xsin⁡x+cos⁡x=sin⁡x+cos⁡x\frac{1+2 \sin x \cos x}{\sin x+\cos x} = \sin x+\cos x

Our goal is to solve this equation and uncover the values of x that satisfy it. We will employ various trigonometric identities and techniques to simplify the equation and ultimately arrive at a solution.

Understanding the Equation

Before we begin solving the equation, let's take a closer look at its structure. The equation involves a fraction with a numerator and a denominator, both of which contain trigonometric functions. The numerator is a sum of two terms, while the denominator is a sum of two terms as well.

To make the equation more manageable, we can start by simplifying the numerator using the double-angle formula for sine:

sin⁡xcos⁡x=12sin⁡2x\sin x \cos x = \frac{1}{2} \sin 2x

Substituting this expression into the numerator, we get:

1+2sin⁡xcos⁡x=1+sin⁡2x1+2 \sin x \cos x = 1 + \sin 2x

Simplifying the Equation

Now that we have simplified the numerator, let's focus on the denominator. We can rewrite the denominator as:

sin⁡x+cos⁡x=2(12sin⁡x+12cos⁡x)\sin x + \cos x = \sqrt{2} \left( \frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x \right)

Using the angle addition formula for sine, we can rewrite the expression inside the parentheses as:

12sin⁡x+12cos⁡x=sin⁡(x+π4)\frac{1}{\sqrt{2}} \sin x + \frac{1}{\sqrt{2}} \cos x = \sin \left( x + \frac{\pi}{4} \right)

Substituting this expression into the denominator, we get:

sin⁡x+cos⁡x=2sin⁡(x+π4)\sin x + \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right)

Equating the Numerator and Denominator

Now that we have simplified the numerator and denominator, let's equate the two expressions:

1+sin⁡2x=2sin⁡(x+π4)1 + \sin 2x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right)

Using Trigonometric Identities

To further simplify the equation, we can use the double-angle formula for sine:

sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x

Substituting this expression into the equation, we get:

1+2sin⁡xcos⁡x=2sin⁡(x+π4)1 + 2 \sin x \cos x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right)

Simplifying the Equation Further

Using the double-angle formula for sine, we can rewrite the equation as:

1+sin⁡2x=2sin⁡(x+π4)1 + \sin 2x = \sqrt{2} \sin \left( x + \frac{\pi}{4} \right)

Solving for x

To solve for x, we can use the inverse sine function:

sin⁡−1(1+sin⁡2x2)=x+π4\sin^{-1} \left( \frac{1 + \sin 2x}{\sqrt{2}} \right) = x + \frac{\pi}{4}

Final Solution

After simplifying the equation and using trigonometric identities, we arrive at the final solution:

x=sin⁡−1(1+sin⁡2x2)−π4x = \sin^{-1} \left( \frac{1 + \sin 2x}{\sqrt{2}} \right) - \frac{\pi}{4}

Conclusion

In this article, we have solved a fascinating trigonometric equation involving sine and cosine functions. We have employed various trigonometric identities and techniques to simplify the equation and ultimately arrive at a solution. The final solution involves the inverse sine function and is expressed in terms of x.

Future Work

In future work, we can explore other trigonometric equations and use similar techniques to simplify and solve them. We can also investigate the properties of the inverse sine function and its applications in various fields.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak
  • [3] "Trigonometric Identities" by Math Open Reference

Acknowledgments

We would like to thank the authors of the references listed above for their contributions to the field of trigonometry and calculus. We would also like to thank our readers for their interest in this article and for their feedback and suggestions.

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Q: What is the main goal of solving the trigonometric equation?

A: The main goal of solving the trigonometric equation is to find the values of x that satisfy the equation.

Q: What are the key steps involved in solving the trigonometric equation?

A: The key steps involved in solving the trigonometric equation are:

  1. Simplifying the numerator using the double-angle formula for sine.
  2. Rewriting the denominator using the angle addition formula for sine.
  3. Equating the numerator and denominator.
  4. Using trigonometric identities to simplify the equation further.
  5. Solving for x using the inverse sine function.

Q: What is the significance of the double-angle formula for sine in solving the trigonometric equation?

A: The double-angle formula for sine is used to simplify the numerator and express it in terms of a single trigonometric function.

Q: How is the angle addition formula for sine used in solving the trigonometric equation?

A: The angle addition formula for sine is used to rewrite the denominator and express it in terms of a single trigonometric function.

Q: What is the role of the inverse sine function in solving the trigonometric equation?

A: The inverse sine function is used to solve for x by expressing the equation in terms of x.

Q: Can the trigonometric equation be solved using other methods?

A: Yes, the trigonometric equation can be solved using other methods such as using the tangent function or the cotangent function.

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A: Trigonometric equations have numerous applications in real-world problems such as:

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A: The trigonometric equation can be modified to solve for other trigonometric functions such as the tangent function or the cotangent function by using the appropriate trigonometric identities.

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

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  • Not using the correct trigonometric identities
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