Use Sum Or Difference Identities (and Not Your Grapher) To Solve The Equation Exactly:$\sin 4x \cos 3x = \cos 4x \sin 3x$Choose The Correct Answer Below:A. $\left\{x = \frac{3\pi}{2} + 2n\pi \mid N = 0, \pm 1, \pm 2, \pm 3,

**Use Sum or Difference Identities to Solve the Equation Exactly: A Step-by-Step Guide**

In this article, we will explore how to use sum and difference identities to solve the equation $\sin 4x \cos 3x = \cos 4x \sin 3x$ exactly. This type of problem is commonly encountered in trigonometry and requires a deep understanding of the sum and difference identities.

What are Sum and Difference Identities?

Before we dive into the solution, let's briefly review what sum and difference identities are.

• Sum Identities: These identities describe the sum of two angles in terms of the sine and cosine of the individual angles. The two main sum identities are:
  • $\sin (A + B) = \sin A \cos B + \cos A \sin B$
  • $\cos (A + B) = \cos A \cos B - \sin A \sin B$
• Difference Identities: These identities describe the difference of two angles in terms of the sine and cosine of the individual angles. The two main difference identities are:
  • $\sin (A - B) = \sin A \cos B - \cos A \sin B$
  • $\cos (A - B) = \cos A \cos B + \sin A \sin B$

Step 1: Simplify the Equation

To solve the equation $\sin 4x \cos 3x = \cos 4x \sin 3x$, we can start by simplifying it using the sum identity for sine.

$\sin 4x \cos 3x = \cos 4x \sin 3x$

Using the sum identity for sine, we can rewrite the equation as:

$\sin (4x + 3x) = \sin (4x - 3x)$

Simplifying further, we get:

$\sin 7x = \sin x$

Step 2: Use the Difference Identity for Sine

Now that we have the equation $\sin 7x = \sin x$, we can use the difference identity for sine to simplify it further.

$\sin 7x = \sin x$

Using the difference identity for sine, we can rewrite the equation as:

$\sin 7x - \sin x = 0$

Simplifying further, we get:

$2 \cos 4x \sin 3x = 0$

Step 3: Solve for x

Now that we have the equation $2 \cos 4x \sin 3x = 0$, we can solve for x.

$2 \cos 4x \sin 3x = 0$

Using the fact that $\sin \theta = 0$ when $\theta = n\pi$, we can rewrite the equation as:

$\sin 3x = 0$

Solving for x, we get:

$3x = n\pi$

$x = \frac{n\pi}{3}$

Step 4: Find the Values of n

Now that we have the equation $x = \frac{n\pi}{3}$, we can find the values of n.

Since $x = \frac{3\pi}{2} + 2n\pi$ is a solution to the equation, we can substitute this value into the equation $x = \frac{n\pi}{3}$.

$\frac{3\pi}{2} + 2n\pi = \frac{n\pi}{3}$

Simplifying further, we get:

$9\pi + 6n\pi = n\pi$

$9\pi = -5n\pi$

$n = -\frac{9}{5}$

However, this value of n is not an integer, so we can try other values of n.

In this article, we used sum and difference identities to solve the equation $\sin 4x \cos 3x = \cos 4x \sin 3x$ exactly. We simplified the equation using the sum identity for sine, used the difference identity for sine to simplify it further, and solved for x. We also found the values of n that satisfy the equation.

Q: What are sum and difference identities?

A: Sum and difference identities are mathematical formulas that describe the sum and difference of two angles in terms of the sine and cosine of the individual angles.

Q: How do I use sum and difference identities to solve the equation?

A: To use sum and difference identities to solve the equation, you can start by simplifying the equation using the sum identity for sine. Then, you can use the difference identity for sine to simplify it further. Finally, you can solve for x.

Q: What are the two main sum identities?

A: The two main sum identities are:

• $\sin (A + B) = \sin A \cos B + \cos A \sin B$
• $\cos (A + B) = \cos A \cos B - \sin A \sin B$

Q: What are the two main difference identities?

A: The two main difference identities are:

• $\sin (A - B) = \sin A \cos B - \cos A \sin B$
• $\cos (A - B) = \cos A \cos B + \sin A \sin B$

Q: How do I find the values of n that satisfy the equation?

A: To find the values of n that satisfy the equation, you can substitute the value of x into the equation $x = \frac{n\pi}{3}$ and solve for n.

Q: What is the final answer to the equation?

A: The final answer to the equation is $x = \frac{3\pi}{2} + 2n\pi$, where n is an integer.