Solve the equation for $$ x $$, where $$ x $$ is in radians. State the exact solutions, if any, that lie in the interval $$ [0,2 \pi) $$. If the equation has no solutions, respond with DNE. $$ \cos (4 x) \cos (x)=-\sin (4 x) \sin (x) $$ in the interval $$ [0,2 \pi) $$. If the equation has no solutions, respond DNE. Separate muitiple solutions with a comma.

Question

UpStudy AI · Accepted Answer

The given equation is:

$$
\cos(4x) \cos(x) = -\sin(4x) \sin(x)
$$

We can rewrite the equation by adding both sides:

$$
\cos(4x) \cos(x) + \sin(4x) \sin(x) = 0
$$

Using the cosine angle subtraction identity:

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

we get:

$$
\cos(4x - x) = \cos(3x) = 0
$$

To solve $$\cos(3x) = 0$$, we know that:

$$
3x = \frac{\pi}{2} + k\pi \quad \text{for integer } k
$$

Thus,

$$
x = \frac{\pi}{6} + \frac{k\pi}{3}
$$

We need to find all solutions $$x$$ within the interval $$[0, 2\pi)$$.

By substituting integer values for $$k$$, we get the solutions:

$$
x = \frac{\pi}{6},\ \frac{\pi}{2},\ \frac{5\pi}{6},\ \frac{7\pi}{6},\ \frac{3\pi}{2},\ \frac{11\pi}{6}
$$

**Final Answer:**

$$\dfrac{\pi}{6}, \dfrac{\pi}{2}, \dfrac{5\,\pi}{6}, \dfrac{7\,\pi}{6}, \dfrac{3\,\pi}{2}, \dfrac{11\,\pi}{6}$$
The solutions are $$ \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} $$.

Solve the equation for , where is in radians. State the exact solutions, if any, that lie in the interval
. If the equation has no solutions, respond with DNE.
in the interval . If the equation has no solutions, respond
DNE.
Separate muitiple solutions with a comma.

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Solve the equation for , where is in radians. State the exact solutions, if any, that lie in the interval . If the equation has no solutions, respond with DNE. in the interval . If the equation has no solutions, respond DNE. Separate muitiple solutions with a comma.