4.2 Solve for $$ x $$ $$ 4.2 .1 \frac{\cos (x}{} $$ $$ 4.2 .2 an x $$

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4.2 Solve for $$ x $$ $$ 4.2 .1 \frac{\cos (x}{} $$ $$ 4.2 .2 	an x $$

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It appears that there might be some formatting issues with the equations you've provided. To assist you effectively, I'll make an educated guess about what you're asking and provide solutions for two common types of trigonometric equations:

1. **Solving for $$ x $$ in an equation involving cosine:**
   
   Let's assume the equation is:
   $$
   \cos(x) = a
   $$
   where $$ a $$ is a known value.

2. **Solving for $$ x $$ in an equation involving tangent:**
   
   Let's assume the equation is:
   $$
   \tan(x) = b
   $$
   where $$ b $$ is a known value.

If your equations differ, please provide the correct expressions, and I'll be happy to help!

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### 4.2.1 Solving $$ \cos(x) = a $$

**Step 1: Check the Validity of $$ a $$**
   
The range of the cosine function is $$[-1, 1]$$. Ensure that $$ a $$ falls within this interval. If $$ |a| > 1 $$, there are no real solutions.

**Step 2: Find the General Solution**

If $$ \cos(x) = a $$, then the general solutions are:
$$
x = \pm \arccos(a) + 2\pi k \quad \text{for any integer } k
$$

**Example:**

Solve $$ \cos(x) = \frac{1}{2} $$.

**Solution:**
$$
x = \pm \arccos\left(\frac{1}{2}\right) + 2\pi k = \pm \frac{\pi}{3} + 2\pi k \quad \text{for any integer } k
$$

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### 4.2.2 Solving $$ \tan(x) = b $$

**Step 1: Understand the Tangent Function**
   
The tangent function has a period of $$ \pi $$, and it is defined for all real numbers except where $$ \cos(x) = 0 $$.

**Step 2: Find the General Solution**

If $$ \tan(x) = b $$, then the general solutions are:
$$
x = \arctan(b) + \pi k \quad \text{for any integer } k
$$

**Example:**

Solve $$ \tan(x) = \sqrt{3} $$.

**Solution:**
$$
x = \arctan(\sqrt{3}) + \pi k = \frac{\pi}{3} + \pi k \quad \text{for any integer } k
$$

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### Summary

- **For $$ \cos(x) = a $$:**
  $$
  x = \pm \arccos(a) + 2\pi k \quad \text{for any integer } k
  $$
  
- **For $$ \tan(x) = b $$:**
  $$
  x = \arctan(b) + \pi k \quad \text{for any integer } k
  $$

**Note:** Always specify the interval for $$ x $$ if you're looking for solutions within a particular range, such as $$ 0 \leq x < 2\pi $$.

If you had different equations in mind, please provide the corrected versions, and I'll be glad to help further!
To solve for $$ x $$ in the equations: 1. $$ \cos(x) = a $$: $$ x = \pm \arccos(a) + 2\pi k \quad \text{for any integer } k $$ 2. $$ \tan(x) = b $$: $$ x = \arctan(b) + \pi k \quad \text{for any integer } k $$ Ensure $$ |a| \leq 1 $$ for real solutions.

4.2 Solve for

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