Solve For X X X In The Equation Sin ⁡ ( 2 X ) − Sin ⁡ ( X ) = 0 \sin(2x) - \sin(x) = 0 Sin ( 2 X ) − Sin ( X ) = 0 , Where 0 ≤ X \textless 2 Π 0 \leq X \ \textless \ 2\pi 0 ≤ X \textless 2 Π . What Are The Possible Solutions For X X X ?A. { Π 6 , 5 Π 6 } \left\{\frac{\pi}{6}, \frac{5\pi}{6}\right\} { 6 Π ​ , 6 5 Π ​ } B.

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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 $\sin(2x) - \sin(x) = 0$, where $0 \leq x \ \textless \ 2\pi$. This equation is a classic example of a trigonometric equation that can be solved using various techniques, including the sum-to-product identity and the double-angle formula.

Understanding the Equation

The given equation is $\sin(2x) - \sin(x) = 0$. To solve this equation, we need to find the values of $x$ that satisfy the equation. The equation involves the sine function, which is a periodic function with a period of $2\pi$. This means that the sine function repeats its values every $2\pi$ radians.

Using the Sum-to-Product Identity

One of the most useful techniques for solving trigonometric equations is the sum-to-product identity. The sum-to-product identity states that $\sin(a) - \sin(b) = 2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$. We can use this identity to rewrite the given equation as:

$$\sin(2x) - \sin(x) = 2\cos\left(\frac{2x+x}{2}\right)\sin\left(\frac{2x-x}{2}\right)$$

Simplifying the expression, we get:

$$\sin(2x) - \sin(x) = 2\cos\left(\frac{3x}{2}\right)\sin\left(\frac{x}{2}\right)$$

Using the Double-Angle Formula

Another useful technique for solving trigonometric equations is the double-angle formula. The double-angle formula states that $\sin(2a) = 2\sin(a)\cos(a)$. We can use this formula to rewrite the given equation as:

$$\sin(2x) - \sin(x) = 2\sin(x)\cos(x) - \sin(x)$$

Simplifying the expression, we get:

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

Solving the Equation

Now that we have rewritten the equation using the sum-to-product identity and the double-angle formula, we can solve for $x$. Setting the expression equal to zero, we get:

$$\sin(x)(2\cos(x) - 1) = 0$$

This equation has two possible solutions: $\sin(x) = 0$ and $2\cos(x) - 1 = 0$. Solving the first equation, we get:

$$\sin(x) = 0 \Rightarrow x = 0, \pi, 2\pi$$

Solving the second equation, we get:

$$2\cos(x) - 1 = 0 \Rightarrow \cos(x) = \frac{1}{2} \Rightarrow x = \frac{\pi}{3}, \frac{5\pi}{3}$$

Finding the Final Solutions

Now that we have found the possible solutions for $x$, we need to check if they satisfy the original equation. Plugging in the values of $x$ into the original equation, we get:

$$\sin(2x) - \sin(x) = 0$$

Evaluating the expression, we get:

$$\sin(2x) - \sin(x) = 0 \Rightarrow \sin\left(2\cdot\frac{\pi}{6}\right) - \sin\left(\frac{\pi}{6}\right) = 0$$

$$\sin\left(2\cdot\frac{\pi}{6}\right) - \sin\left(\frac{\pi}{6}\right) = \sin\left(\frac{\pi}{3}\right) - \sin\left(\frac{\pi}{6}\right) = 0$$

Similarly, we can evaluate the expression for the other possible solutions:

$$\sin(2x) - \sin(x) = 0 \Rightarrow \sin\left(2\cdot\frac{5\pi}{6}\right) - \sin\left(\frac{5\pi}{6}\right) = 0$$

$$\sin\left(2\cdot\frac{5\pi}{6}\right) - \sin\left(\frac{5\pi}{6}\right) = \sin\left(\frac{5\pi}{3}\right) - \sin\left(\frac{5\pi}{6}\right) = 0$$

$$\sin(2x) - \sin(x) = 0 \Rightarrow \sin\left(2\cdot\pi\right) - \sin\left(\pi\right) = 0$$

$$\sin\left(2\cdot\pi\right) - \sin\left(\pi\right) = \sin(0) - \sin(\pi) = 0$$

Conclusion

In this article, we have solved the equation $\sin(2x) - \sin(x) = 0$, where $0 \leq x \ \textless \ 2\pi$. We have used the sum-to-product identity and the double-angle formula to rewrite the equation and solve for $x$. The possible solutions for $x$ are $\left\{\frac{\pi}{6}, \frac{5\pi}{6}\right\}$.

Final Answer

The final answer is $\boxed{\left\{\frac{\pi}{6}, \frac{5\pi}{6}\right\}}$.

Introduction

In our previous article, we solved the equation $\sin(2x) - \sin(x) = 0$, where $0 \leq x \ \textless \ 2\pi$. We used the sum-to-product identity and the double-angle formula to rewrite the equation and solve for $x$. In this article, we will provide a Q&A guide to help you understand the solution and apply it to other trigonometric equations.

Q: What is the sum-to-product identity?

A: The sum-to-product identity is a formula that allows us to rewrite the difference of two sine functions as a product of two functions. It is given by:

$$\sin(a) - \sin(b) = 2\cos\left(\frac{a+b}{2}\right)\sin\left(\frac{a-b}{2}\right)$$

Q: How do I use the sum-to-product identity to solve trigonometric equations?

A: To use the sum-to-product identity to solve trigonometric equations, you need to identify the two sine functions in the equation and rewrite them using the sum-to-product identity. Then, you can simplify the expression and solve for the variable.

Q: What is the double-angle formula?

A: The double-angle formula is a formula that allows us to rewrite the sine of a double angle as a product of two sine functions. It is given by:

$$\sin(2a) = 2\sin(a)\cos(a)$$

Q: How do I use the double-angle formula to solve trigonometric equations?

A: To use the double-angle formula to solve trigonometric equations, you need to identify the double angle in the equation and rewrite it using the double-angle formula. Then, you can simplify the expression and solve for the variable.

Q: What are some common trigonometric equations that can be solved using the sum-to-product identity and the double-angle formula?

A: Some common trigonometric equations that can be solved using the sum-to-product identity and the double-angle formula include:

• $\sin(a) - \sin(b) = 0$
• $\cos(a) - \cos(b) = 0$
• $\sin(2a) - \sin(a) = 0$
• $\cos(2a) - \cos(a) = 0$

Q: How do I check if a solution satisfies the original equation?

A: To check if a solution satisfies the original equation, you need to plug the solution into the original equation and evaluate the expression. If the expression is equal to zero, then the solution satisfies the original equation.

Q: What are some tips for solving trigonometric equations?

A: Some tips for solving trigonometric equations include:

• Use the sum-to-product identity and the double-angle formula to rewrite the equation.
• Simplify the expression and solve for the variable.
• Check if the solution satisfies the original equation.
• Use trigonometric identities to rewrite the equation and simplify the expression.

Conclusion

In this article, we have provided a Q&A guide to help you understand the solution to the equation $\sin(2x) - \sin(x) = 0$, where $0 \leq x \ \textless \ 2\pi$. We have also provided some tips for solving trigonometric equations and some common trigonometric equations that can be solved using the sum-to-product identity and the double-angle formula.

Final Answer

The final answer is $\boxed{\left\{\frac{\pi}{6}, \frac{5\pi}{6}\right\}}$.