Solve The Equation:$3 - \sin \theta = \cos (2 \theta$\]

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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 the equation $3 - \sin \theta = \cos (2 \theta)$, which involves a combination of trigonometric functions and identities. We will break down the solution into manageable steps, making it easier for readers to understand and follow along.

Understanding the Equation

The given equation is $3 - \sin \theta = \cos (2 \theta)$. To solve this equation, we need to use trigonometric identities and properties to simplify and manipulate the equation. The first step is to recognize that $\cos (2 \theta)$ can be expressed in terms of $\sin \theta$ using the double-angle identity.

Double-Angle Identity

The double-angle identity for cosine is given by:

$$\cos (2 \theta) = 1 - 2 \sin^2 \theta$$

We can substitute this expression into the original equation to get:

$$3 - \sin \theta = 1 - 2 \sin^2 \theta$$

Simplifying the Equation

Now, we can simplify the equation by combining like terms:

$$2 = 2 \sin^2 \theta - \sin \theta$$

Rearranging the Equation

To make it easier to solve, we can rearrange the equation to get:

$$2 \sin^2 \theta - \sin \theta - 2 = 0$$

Factoring the Quadratic

The equation is a quadratic in terms of $\sin \theta$, and we can factor it as:

$$(2 \sin \theta + 1)(\sin \theta - 2) = 0$$

Solving for $\sin \theta$

Now, we can set each factor equal to zero and solve for $\sin \theta$:

$$2 \sin \theta + 1 = 0 \Rightarrow \sin \theta = -\frac{1}{2}$$

$$\sin \theta - 2 = 0 \Rightarrow \sin \theta = 2$$

Checking the Solutions

However, we need to check if these solutions are valid. Since $\sin \theta$ cannot be greater than 1, the solution $\sin \theta = 2$ is not valid.

Valid Solution

The only valid solution is $\sin \theta = -\frac{1}{2}$.

Finding the Value of $\theta$

To find the value of $\theta$, we can use the inverse sine function:

$$\theta = \sin^{-1} \left(-\frac{1}{2}\right)$$

Evaluating the Inverse Sine

The inverse sine of $-\frac{1}{2}$ is:

$$\theta = -\frac{\pi}{6}$$

Conclusion

In this article, we solved the trigonometric equation $3 - \sin \theta = \cos (2 \theta)$ using a combination of trigonometric identities and properties. We broke down the solution into manageable steps, making it easier for readers to understand and follow along. The final solution is $\sin \theta = -\frac{1}{2}$, and the corresponding value of $\theta$ is $-\frac{\pi}{6}$.

Common Trigonometric Identities

Here are some common trigonometric identities that are useful for solving trigonometric equations:

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Double-Angle Identity: $\cos (2 \theta) = 1 - 2 \sin^2 \theta$
• Half-Angle Identity: $\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$
• Sum and Difference Identities: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• Product-to-Sum Identities: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Tips for Solving Trigonometric Equations

Here are some tips for solving trigonometric equations:

• Use Trigonometric Identities: Trigonometric identities can help simplify and manipulate trigonometric equations.
• Check for Extraneous Solutions: Be sure to check for extraneous solutions, as some solutions may not be valid.
• Use Inverse Trigonometric Functions: Inverse trigonometric functions can help find the value of $\theta$.
• Graphing: Graphing can help visualize the solutions to a trigonometric equation.

Real-World Applications

Trigonometric equations have many real-world applications, including:

• Physics: Trigonometric equations are used to describe the motion of objects in physics.
• Engineering: Trigonometric equations are used to design and analyze engineering systems.
• Navigation: Trigonometric equations are used in navigation to determine the position and orientation of objects.
• Computer Science: Trigonometric equations are used in computer science to solve problems in graphics, game development, and more.

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and identities. By using a combination of trigonometric identities and properties, we can simplify and manipulate trigonometric equations to find their solutions. The final solution to the equation $3 - \sin \theta = \cos (2 \theta)$ is $\sin \theta = -\frac{1}{2}$, and the corresponding value of $\theta$ is $-\frac{\pi}{6}$.

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 problems and can be solved using various techniques.

Q: What are some common trigonometric identities?

A: Some common trigonometric identities include:

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Double-Angle Identity: $\cos (2 \theta) = 1 - 2 \sin^2 \theta$
• Half-Angle Identity: $\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$
• Sum and Difference Identities: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• Product-to-Sum Identities: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Q: How do I solve a trigonometric equation?

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

1. Simplify the equation: Use trigonometric identities to simplify the equation.
2. Isolate the trigonometric function: Isolate the trigonometric function on one side of the equation.
3. Use inverse trigonometric functions: Use inverse trigonometric functions to find the value of the trigonometric function.
4. Check for extraneous solutions: Check for extraneous solutions, as some solutions may not be valid.

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

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

Q: Can trigonometric equations be used to model real-world problems?

A: Yes, trigonometric equations can be used to model real-world problems. For example, trigonometric equations can be used to model the motion of objects in physics, the design of engineering systems, and the navigation of objects.

Q: What are some common applications of trigonometric equations?

A: Some common applications of trigonometric equations include:

• Physics: Trigonometric equations are used to describe the motion of objects in physics.
• Engineering: Trigonometric equations are used to design and analyze engineering systems.
• Navigation: Trigonometric equations are used in navigation to determine the position and orientation of objects.
• Computer Science: Trigonometric equations are used in computer science to solve problems in graphics, game development, and more.

Q: How do I graph a trigonometric equation?

A: To graph a trigonometric equation, follow these steps:

1. Identify the trigonometric function: Identify the trigonometric function in the equation.
2. Determine the amplitude: Determine the amplitude of the trigonometric function.
3. Determine the period: Determine the period of the trigonometric function.
4. Graph the function: Graph the function using a graphing calculator or software.

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

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

• Not checking for extraneous solutions: Not checking for extraneous solutions can lead to incorrect solutions.
• Not using inverse trigonometric functions: Not using inverse trigonometric functions can make it difficult to find the value of the trigonometric function.
• Not simplifying the equation: Not simplifying the equation can make it difficult to solve.

Q: How do I choose the correct trigonometric identity to use?

A: To choose the correct trigonometric identity to use, follow these steps:

1. Identify the trigonometric function: Identify the trigonometric function in the equation.
2. Determine the type of identity: Determine the type of identity that is needed to simplify the equation.
3. Choose the correct identity: Choose the correct identity to use.

Q: What are some common trigonometric identities that are used in calculus?

A: Some common trigonometric identities that are used in calculus include:

• Pythagorean Identity: $\sin^2 \theta + \cos^2 \theta = 1$
• Double-Angle Identity: $\cos (2 \theta) = 1 - 2 \sin^2 \theta$
• Half-Angle Identity: $\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}$
• Sum and Difference Identities: $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• Product-to-Sum Identities: $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$

Q: How do I use trigonometric identities to simplify a trigonometric equation?

A: To use trigonometric identities to simplify a trigonometric equation, follow these steps:

1. Identify the trigonometric function: Identify the trigonometric function in the equation.
2. Determine the type of identity: Determine the type of identity that is needed to simplify the equation.
3. Choose the correct identity: Choose the correct identity to use.
4. Simplify the equation: Simplify the equation using the chosen identity.

Q: What are some common applications of trigonometric identities in real-world problems?

A: Some common applications of trigonometric identities in real-world problems include:

• Physics: Trigonometric identities are used to describe the motion of objects in physics.
• Engineering: Trigonometric identities are used to design and analyze engineering systems.
• Navigation: Trigonometric identities are used in navigation to determine the position and orientation of objects.
• Computer Science: Trigonometric identities are used in computer science to solve problems in graphics, game development, and more.

Q: How do I use trigonometric identities to solve a trigonometric equation?

A: To use trigonometric identities to solve a trigonometric equation, follow these steps:

1. Identify the trigonometric function: Identify the trigonometric function in the equation.
2. Determine the type of identity: Determine the type of identity that is needed to simplify the equation.
3. Choose the correct identity: Choose the correct identity to use.
4. Simplify the equation: Simplify the equation using the chosen identity.
5. Use inverse trigonometric functions: Use inverse trigonometric functions to find the value of the trigonometric function.
6. Check for extraneous solutions: Check for extraneous solutions, as some solutions may not be valid.