solve the following equations over the specified domain. Yourfinal answers must be exact and simplified (no approximations). **Problem Statement:**b. \(\sqrt{3} \tan(7\theta) = -1\) where \(\theta\) is any real number and \(k\) is any integer. **Discussion:**The given equation involves trigonometric functions and seeks to find values for \(\theta\) that satisfy the equation for any integer \(k\). The challenge is to solve it using trigonometric identities and properties of the tangent function. **Approach:**1. **Isolate the Tangent Function:** \[ \tan(7\theta) = -\frac{1}{\sqrt{3}} \]2. **Consider the Special Angles:** Recognize that \(\tan(\theta) = -\frac{1}{\sqrt{3}}\) corresponds to angles where the tangent function has this specific value. Therefore, consider the angles in the unit circle where this occurs, keeping in mind the periodic nature of the tangent function: \[ 7\theta = \left\{-\frac{\pi}{6} + k\pi\right\} \]3. **Solve for \(\theta\):** Divide by 7 to isolate \(\theta\): \[ \theta = \left\{-\frac{\pi}{42} + \frac{k\pi}{7}\right\} \]Hence, the general solution involves considering all real numbers \(\theta\) and integers \(k\).

Name: solve the following equations over the specified domain. Yourfinal ans
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: solve the following equations over the specified domain. Yourfinal answers must be exact and simplified (no approximations). **Problem Statement:**b. \(\sqrt{3} \tan(7\theta) = -1\) where \(\theta\) is any real number and \(k\) is any integer. **Disc