What are the solutions of $$ \cos 4 heta=\cos 2 heta=\sin 30 $$ ? Seleet It thater(s) $$ \frac{7 \pi}{6}+2 \pi \pi $$ $$ \frac{n \pi}{4} $$ $$ \frac{7 \pi}{6}+n \pi $$ $$ \frac{\pi}{4}+n \pi $$ $$ 0+\frac{n \pi}{3} $$ $$ \frac{11 \pi}{6}+m \pi $$ $$ \frac{3 \pi}{4}+n \pi $$ $$ \frac{11 \pi}{6}+2 n \pi $$

Question

What are the solutions of $$ \cos 4 	heta=\cos 2 	heta=\sin 30 $$ ? Seleet It thater(s) $$ \frac{7 \pi}{6}+2 \pi \pi $$ $$ \frac{n \pi}{4} $$ $$ \frac{7 \pi}{6}+n \pi $$ $$ \frac{\pi}{4}+n \pi $$ $$ 0+\frac{n \pi}{3} $$ $$ \frac{11 \pi}{6}+m \pi $$ $$ \frac{3 \pi}{4}+n \pi $$ $$ \frac{11 \pi}{6}+2 n \pi $$

UpStudy AI · Accepted Answer

To solve the equation $$ \cos 4\theta = \cos 2\theta = \sin 30^\circ $$, we first need to evaluate $$ \sin 30^\circ $$:

$$
\sin 30^\circ = \frac{1}{2}
$$

Now we can rewrite the equations as:

1. $$ \cos 4\theta = \frac{1}{2} $$
2. $$ \cos 2\theta = \frac{1}{2} $$

### Step 1: Solve $$ \cos 4\theta = \frac{1}{2} $$

The general solutions for $$ \cos x = \frac{1}{2} $$ are:

$$
x = 2n\pi \pm \frac{\pi}{3} \quad (n \in \mathbb{Z})
$$

Substituting $$ x = 4\theta $$:

$$
4\theta = 2n\pi \pm \frac{\pi}{3}
$$

Dividing by 4:

$$
\theta = \frac{n\pi}{2} \pm \frac{\pi}{12}
$$

### Step 2: Solve $$ \cos 2\theta = \frac{1}{2} $$

Similarly, for $$ \cos x = \frac{1}{2} $$:

$$
x = 2m\pi \pm \frac{\pi}{3} \quad (m \in \mathbb{Z})
$$

Substituting $$ x = 2\theta $$:

$$
2\theta = 2m\pi \pm \frac{\pi}{3}
$$

Dividing by 2:

$$
\theta = m\pi \pm \frac{\pi}{6}
$$

### Step 3: Combine the solutions

Now we have two sets of solutions:

1. From $$ \cos 4\theta = \frac{1}{2} $$:
   $$
   \theta = \frac{n\pi}{2} + \frac{\pi}{12} \quad \text{and} \quad \theta = \frac{n\pi}{2} - \frac{\pi}{12}
   $$

2. From $$ \cos 2\theta = \frac{1}{2} $$:
   $$
   \theta = m\pi + \frac{\pi}{6} \quad \text{and} \quad \theta = m\pi - \frac{\pi}{6}
   $$

### Step 4: Identify the solutions from the options

Now we will check which of the provided options match the solutions we derived.

1. $$ \frac{7\pi}{6} + 2\pi n $$
2. $$ \frac{n\pi}{4} $$
3. $$ \frac{7\pi}{6} + n\pi $$
4. $$ \frac{\pi}{4} + n\pi $$
5. $$ 0 + \frac{n\pi}{3} $$
6. $$ \frac{11\pi}{6} + m\pi $$
7. $$ \frac{3\pi}{4} + n\pi $$
8. $$ \frac{11\pi}{6} + 2n\pi $$

### Step 5: Check for specific values

- For $$ \theta = \frac{n\pi}{2} + \frac{\pi}{12} $$ and $$ \theta = \frac{n\pi}{2} - \frac{\pi}{12} $$, we can find specific values of $$ n $$ that yield results in the options.
- For $$ \theta = m\pi + \frac{\pi}{6} $$ and $$ \theta = m\pi - \frac{\pi}{6} $$, we can also find specific values of $$ m $$.

After checking the options, we find that:

- $$ \frac{7\pi}{6} + n\pi $$ can be derived from $$ \theta = m\pi + \frac{\pi}{6} $$ for specific values of $$ m $$.
- $$ \frac{11\pi}{6} + m\pi $$ can also be derived from the same.

Thus, the solutions that match the derived equations are:

- $$ \frac{7\pi}{6} + n\pi $$
- $$ \frac{11\pi}{6} + m\pi $$

These are the solutions to the given problem.
The solutions are $$ \frac{7\pi}{6} + n\pi $$ and $$ \frac{11\pi}{6} + m\pi $$.

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