Solve For Angle Θ \theta Θ . Sin ⁡ ( Θ ) + Sin ⁡ ( Θ ) Cos ⁡ ( Θ ) = 0 \sin(\theta) + \sin(\theta) \cos(\theta) = 0 Sin ( Θ ) + Sin ( Θ ) Cos ( Θ ) = 0 A. Θ = N Π \theta = N\pi Θ = Nπ , Where N N N Is An Integer B. Θ = Π 3 + 2 N Π \theta = \frac{\pi}{3} + 2n\pi Θ = 3 Π ​ + 2 Nπ , Where N N N Is An Integer C. $\theta =

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 a specific trigonometric equation involving sine and cosine functions. We will break down the solution step by step, providing a clear and concise explanation of each step.

The Given Equation

The given equation is:

$\sin(\theta) + \sin(\theta) \cos(\theta) = 0$

Our goal is to solve for the angle $\theta$.

Step 1: Factor Out the Common Term

The first step in solving this equation is to factor out the common term $\sin(\theta)$ from the left-hand side of the equation.

$\sin(\theta) (1 + \cos(\theta)) = 0$

Step 2: Apply the Zero Product Property

The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. Applying this property to our equation, we get:

$\sin(\theta) = 0$ or $1 + \cos(\theta) = 0$

Step 3: Solve the First Equation

The first equation is $\sin(\theta) = 0$. We know that the sine function is equal to zero at integer multiples of $\pi$. Therefore, we can write:

$\sin(\theta) = 0 \Rightarrow \theta = n\pi$, where $n$ is an integer

Step 4: Solve the Second Equation

The second equation is $1 + \cos(\theta) = 0$. We can rewrite this equation as:

$\cos(\theta) = -1$

We know that the cosine function is equal to -1 at odd multiples of $\frac{\pi}{2}$. Therefore, we can write:

$\cos(\theta) = -1 \Rightarrow \theta = \frac{\pi}{2} + 2n\pi$, where $n$ is an integer

Step 5: Combine the Solutions

We have found two possible solutions for the angle $\theta$: $\theta = n\pi$ and $\theta = \frac{\pi}{2} + 2n\pi$. However, we need to check if these solutions satisfy the original equation.

Checking the Solutions

Let's substitute the first solution, $\theta = n\pi$, into the original equation:

$\sin(n\pi) + \sin(n\pi) \cos(n\pi) = 0$

Since $\sin(n\pi) = 0$ and $\cos(n\pi) = 1$, we get:

$0 + 0 \cdot 1 = 0$

This solution satisfies the original equation.

Now, let's substitute the second solution, $\theta = \frac{\pi}{2} + 2n\pi$, into the original equation:

$\sin(\frac{\pi}{2} + 2n\pi) + \sin(\frac{\pi}{2} + 2n\pi) \cos(\frac{\pi}{2} + 2n\pi) = 0$

Since $\sin(\frac{\pi}{2} + 2n\pi) = 1$ and $\cos(\frac{\pi}{2} + 2n\pi) = 0$, we get:

$1 + 1 \cdot 0 = 1$

This solution does not satisfy the original equation.

Conclusion

In conclusion, the solutions to the given trigonometric equation are:

$\theta = n\pi$, where $n$ is an integer

Therefore, the correct answer is:

A. $\theta = n\pi$, where $n$ is an integer

Discussion

This problem requires a deep understanding of trigonometric functions and their properties. The solution involves factoring out a common term, applying the zero product property, and solving two separate equations. The final answer is a simple expression involving the integer $n$. This problem is a great example of how trigonometric equations can be solved using algebraic techniques.

Additional Resources

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

• Trigonometric Equations
• Solving Trigonometric Equations
• Trigonometric Functions

Final Answer

The final answer is:

$$$$

Introduction

In our previous article, we solved a trigonometric equation involving sine and cosine functions. In this article, we will provide a Q&A guide to help you better understand the solution and apply it to similar problems.

Q: What is the zero product property?

A: The zero product property states that if the product of two or more factors is equal to zero, then at least one of the factors must be equal to zero. This property is used to solve equations of the form $ab = 0$, where $a$ and $b$ are expressions.

Q: How do I apply the zero product property to trigonometric equations?

A: To apply the zero product property to trigonometric equations, you need to factor out the common term from the left-hand side of the equation. Then, you can set each factor equal to zero and solve for the variable.

Q: What are the common terms in trigonometric equations?

A: The common terms in trigonometric equations are the sine and cosine functions. You can factor out the sine function from the left-hand side of the equation, or the cosine function, depending on the equation.

Q: How do I solve the equation $\sin(\theta) + \sin(\theta) \cos(\theta) = 0$?

A: To solve the equation $\sin(\theta) + \sin(\theta) \cos(\theta) = 0$, you need to factor out the common term $\sin(\theta)$ from the left-hand side of the equation. Then, you can set each factor equal to zero and solve for the variable.

Q: What are the solutions to the equation $\sin(\theta) + \sin(\theta) \cos(\theta) = 0$?

A: The solutions to the equation $\sin(\theta) + \sin(\theta) \cos(\theta) = 0$ are:

$\theta = n\pi$, where $n$ is an integer

Q: How do I check if the solutions satisfy the original equation?

A: To check if the solutions satisfy the original equation, you need to substitute the solution into the original equation and simplify. If the result is equal to zero, then the solution satisfies the original equation.

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

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

• Not factoring out the common term
• Not applying the zero product property
• Not checking if the solutions satisfy the original equation
• Not using the correct trigonometric identities

Q: How can I practice solving trigonometric equations?

A: You can practice solving trigonometric equations by working through examples and exercises in your textbook or online resources. You can also try solving trigonometric equations on your own and checking your solutions with a calculator or online tool.

Conclusion

In conclusion, solving trigonometric equations requires a deep understanding of trigonometric functions and their properties. By applying the zero product property and factoring out common terms, you can solve equations involving sine and cosine functions. Remember to check if the solutions satisfy the original equation and avoid common mistakes when solving trigonometric equations.

Additional Resources

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

• Trigonometric Equations
• Solving Trigonometric Equations
• Trigonometric Functions

Final Answer

The final answer is:

A. $\theta = n\pi$, where $n$ is an integer