Solve For $x$.$12 \sin^2 X = \sin 2x + \cos^2 X$

Introduction

Trigonometric equations are a fundamental part of mathematics, and solving them requires a deep understanding of trigonometric functions and identities. In this article, we will focus on solving a specific trigonometric equation involving sine and cosine functions. The given equation is:

$$12 \sin^2 x = \sin 2x + \cos^2 x$$

Our goal is to solve for $x$ and understand the underlying concepts that make this equation solvable.

Understanding the Equation

Before we dive into solving the equation, let's break it down and understand its components. The equation involves two trigonometric functions: sine and cosine. The sine function is represented by $\sin x$, and the cosine function is represented by $\cos x$. The equation also involves the double-angle identity for sine, which is $\sin 2x = 2 \sin x \cos x$.

Step 1: Simplify the Equation

To simplify the equation, we can start by using the double-angle identity for sine:

$$12 \sin^2 x = 2 \sin x \cos x + \cos^2 x$$

Next, we can rewrite the equation as:

$$12 \sin^2 x - \cos^2 x = 2 \sin x \cos x$$

Step 2: Use Trigonometric Identities

Now, we can use the Pythagorean identity, which states that $\sin^2 x + \cos^2 x = 1$. We can rewrite the equation as:

$$12 (1 - \cos^2 x) - \cos^2 x = 2 \sin x \cos x$$

Simplifying further, we get:

$$12 - 13 \cos^2 x = 2 \sin x \cos x$$

Step 3: Rearrange the Equation

To make the equation more manageable, we can rearrange it to isolate the cosine term:

$$13 \cos^2 x = 12 - 2 \sin x \cos x$$

Dividing both sides by 13, we get:

$$\cos^2 x = \frac{12 - 2 \sin x \cos x}{13}$$

Step 4: Use the Double-Angle Identity

Now, we can use the double-angle identity for cosine, which is $\cos 2x = 2 \cos^2 x - 1$. We can rewrite the equation as:

$$\cos 2x = 2 \left( \frac{12 - 2 \sin x \cos x}{13} \right) - 1$$

Simplifying further, we get:

$$\cos 2x = \frac{24 - 4 \sin x \cos x}{13} - 1$$

Step 5: Solve for $x$

Now, we can solve for $x$ by using the inverse cosine function:

$$2x = \cos^{-1} \left( \frac{24 - 4 \sin x \cos x}{13} - 1 \right)$$

Dividing both sides by 2, we get:

$$x = \frac{1}{2} \cos^{-1} \left( \frac{24 - 4 \sin x \cos x}{13} - 1 \right)$$

Conclusion

Solving the given trigonometric equation involves several steps, including simplifying the equation, using trigonometric identities, rearranging the equation, and using the inverse cosine function. By following these steps, we can solve for $x$ and understand the underlying concepts that make this equation solvable.

Final Answer

The final answer is:

$$x = \frac{1}{2} \cos^{-1} \left( \frac{24 - 4 \sin x \cos x}{13} - 1 \right)$$

Note

The final answer is a general solution, and there may be multiple values of $x$ that satisfy the equation. To find the specific values of $x$, we need to use numerical methods or graphing tools.

References

• [1] "Trigonometry" by Michael Corral
• [2] "Calculus" by Michael Spivak
• [3] "Trigonometric Equations" by Wolfram MathWorld

Additional Resources

• [1] Khan Academy: Trigonometry
• [2] MIT OpenCourseWare: Trigonometry
• [3] Wolfram Alpha: Trigonometric Equations
  Frequently Asked Questions: Solving Trigonometric Equations ===========================================================

Q: What is a trigonometric equation?

A: A trigonometric equation is an equation that involves trigonometric functions, such as sine, cosine, and tangent. These equations can be used to model real-world phenomena, such as the motion of objects or the behavior of electrical circuits.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• The Pythagorean identity: $\sin^2 x + \cos^2 x = 1$
• The double-angle identity for sine: $\sin 2x = 2 \sin x \cos x$
• The double-angle identity for cosine: $\cos 2x = 2 \cos^2 x - 1$
• The 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 solve a trigonometric equation?

A: To solve a trigonometric equation, you can follow these steps:

1. Simplify the equation by using trigonometric identities.
2. Rearrange the equation to isolate the trigonometric function.
3. Use the inverse trigonometric function to solve for the variable.
4. Check your solution by plugging it back into the original equation.

Q: What is the difference between a trigonometric equation and a trigonometric function?

A: A trigonometric function is a mathematical function that involves trigonometric operations, such as sine, cosine, and tangent. A trigonometric equation, on the other hand, is an equation that involves trigonometric functions.

Q: Can I use trigonometric equations to model real-world phenomena?

A: Yes, trigonometric equations can be used to model real-world phenomena, such as the motion of objects, the behavior of electrical circuits, and the behavior of sound waves.

Q: What are some common applications of trigonometric equations?

A: Some common applications of trigonometric equations include:

• Navigation: Trigonometric equations can be used to calculate distances and angles between objects.
• Physics: Trigonometric equations can be used to model the motion of objects and the behavior of electrical circuits.
• Engineering: Trigonometric equations can be used to design and optimize systems, such as bridges and buildings.
• Computer Science: Trigonometric equations can be used to develop algorithms and models for computer graphics and game development.

Q: How do I choose the right trigonometric identity to use in a problem?

A: To choose the right trigonometric identity, you need to analyze the equation and identify the trigonometric functions involved. Then, you can use the appropriate identity to simplify the equation.

Q: Can I use trigonometric equations to solve problems in other areas of mathematics?

A: Yes, trigonometric equations can be used to solve problems in other areas of mathematics, such as algebra and calculus.

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.
• Not simplifying the equation correctly.
• Not checking the solution by plugging it back into the original equation.

Q: How do I practice solving trigonometric equations?

A: To practice solving trigonometric equations, you can try the following:

• Work on practice problems from a textbook or online resource.
• Use online tools and calculators to check your solutions.
• Join a study group or find a study partner to work on problems together.
• Take online courses or watch video tutorials to learn new skills and techniques.

Conclusion

Solving trigonometric equations requires a deep understanding of trigonometric functions and identities. By following the steps outlined in this article and practicing regularly, you can become proficient in solving trigonometric equations and apply them to real-world problems.