Solving a Trigonometric Equation In Exercises \( \begin{array}{ll}\mathbf{2 5 - 3 8} \text {, find all solutions of the equation in the interval } \\ \mathbf{0 , 2} \pi) \text {. } \\ \begin{array}{ll}\text { 25. } \cos ^{3} x=\cos x & \text { 26. } \sec ^{2} x-1=0 \\ \text { 27. } 3 \tan ^{3} x=\tan x & \text { 28. } 2 \sin ^{2} x=2+\cos x\end{array}\end{array} \)

Certainly! Let's solve each of the trigonometric equations provided within the interval \([0, 2\pi)\). We'll tackle problems **25** through **28** one by one. --- ### **Problem 25:** \( \cos^{3}x = \cos x \) **Step 1:** Rewrite the equation and factor it. \[ \cos^{3}x - \cos x = 0 \quad \Rightarrow \quad \cos x (\cos^{2}x - 1) = 0 \] **Step 2:** Set each factor equal to zero. 1. \( \cos x = 0 \) 2. \( \cos^{2}x - 1 = 0 \) \(\Rightarrow\) \( \cos x = \pm 1 \) **Step 3:** Find all solutions in \([0, 2\pi)\). - \( \cos x = 0 \) at: - \( x = \frac{\pi}{2} \) - \( x = \frac{3\pi}{2} \) - \( \cos x = 1 \) at: - \( x = 0 \) (or \( 2\pi \), but \( 2\pi \) is excluded) - \( \cos x = -1 \) at: - \( x = \pi \) ****Solutions:** \[ x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \] --- ### **Problem 26:** \( \sec^{2}x - 1 = 0 \) **Step 1:** Recall the identity \( \sec^{2}x = 1 + \tan^{2}x \), but directly solve the equation: \[ \sec^{2}x - 1 = 0 \quad \Rightarrow \quad \sec^{2}x = 1 \quad \Rightarrow \quad \tan^{2}x = 0 \quad \Rightarrow \quad \tan x = 0 \] **Step 2:** Find all solutions where \( \tan x = 0 \) in \([0, 2\pi)\). - \( \tan x = 0 \) at: - \( x = 0 \) - \( x = \pi \) ****Solutions:** \[ x = 0, \pi \] --- ### **Problem 27:** \( 3 \tan^{3}x = \tan x \) **Step 1:** Rearrange and factor the equation. \[ 3 \tan^{3}x - \tan x = 0 \quad \Rightarrow \quad \tan x (3 \tan^{2}x - 1) = 0 \] **Step 2:** Set each factor equal to zero. 1. \( \tan x = 0 \) 2. \( 3 \tan^{2}x - 1 = 0 \) \(\Rightarrow\) \( \tan^{2}x = \frac{1}{3} \) \(\Rightarrow\) \( \tan x = \pm \frac{1}{\sqrt{3}} \) **Step 3:** Find all solutions in \([0, 2\pi)\). - \( \tan x = 0 \) at: - \( x = 0 \) - \( x = \pi \) - \( \tan x = \frac{1}{\sqrt{3}} \) corresponds to: - \( x = \frac{\pi}{6} \) - \( x = \frac{7\pi}{6} \) - \( \tan x = -\frac{1}{\sqrt{3}} \) corresponds to: - \( x = \frac{5\pi}{6} \) - \( x = \frac{11\pi}{6} \) ****Solutions:** \[ x = 0, \frac{\pi}{6}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{11\pi}{6} \] --- ### **Problem 28:** \( 2 \sin^{2}x = 2 + \cos x \) **Step 1:** Use the Pythagorean identity \( \sin^{2}x = 1 - \cos^{2}x \) to rewrite the equation. \[ 2(1 - \cos^{2}x) = 2 + \cos x \quad \Rightarrow \quad 2 - 2\cos^{2}x = 2 + \cos x \] **Step 2:** Simplify the equation. \[ -2\cos^{2}x = \cos x \quad \Rightarrow \quad -2\cos^{2}x - \cos x = 0 \quad \Rightarrow \quad \cos x ( -2\cos x - 1 ) = 0 \] **Step 3:** Set each factor equal to zero. 1. \( \cos x = 0 \) 2. \( -2\cos x - 1 = 0 \) \(\Rightarrow\) \( \cos x = -\frac{1}{2} \) **Step 4:** Find all solutions in \([0, 2\pi)\). - \( \cos x = 0 \) at: - \( x = \frac{\pi}{2} \) - \( x = \frac{3\pi}{2} \) - \( \cos x = -\frac{1}{2} \) at: - \( x = \frac{2\pi}{3} \) - \( x = \frac{4\pi}{3} \) ****Solutions:** \[ x = \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{3\pi}{2} \] --- ### **Summary of All Solutions:** - **Problem 25:** \( x = 0, \frac{\pi}{2}, \pi, \frac{3\pi}{2} \) - **Problem 26:** \( x = 0, \pi \) - **Problem 27:** \( x = 0, \frac{\pi}{6}, \frac{5\pi}{6}, \pi, \frac{7\pi}{6}, \frac{11\pi}{6} \) - **Problem 28:** \( x = \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{3\pi}{2} \) Feel free to reach out if you need further clarification on any of these problems!