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

Question

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

UpStudy AI · Accepted Answer

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.

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### **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:**
$$
x = \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{3\pi}{2}
$$

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### **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!
Solutions for the trigonometric equations are as follows: - **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:** $$ x = \frac{\pi}{2}, \frac{2\pi}{3}, \frac{4\pi}{3}, \frac{3\pi}{2} $$

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