Simplify and solve the equation: $$ an(4x) = 1 $$ for all solutions of $$ x $$ in the range $$ (0, 2\pi) $$.

Question

Simplify and solve the equation: $$ 	an(4x) = 1 $$ for all solutions of $$ x $$ in the range $$ (0, 2\pi) $$.

UpStudy AI · Accepted Answer

To solve the equation $$ \tan(4x) = 1 $$ for $$ x $$ in the range $$ (0, 2\pi) $$, follow these steps:

1. **Determine the General Solution for $$ \tan(\theta) = 1 $$:**
   
   The tangent function equals 1 at angles where:
   $$
   \theta = \frac{\pi}{4} + k\pi \quad \text{for any integer } k
   $$
   
2. **Apply It to the Given Equation:**
   
   Set $$ \theta = 4x $$, so:
   $$
   4x = \frac{\pi}{4} + k\pi
   $$
   
   Solving for $$ x $$ gives:
   $$
   x = \frac{\pi}{16} + \frac{k\pi}{4}
   $$
   
3. **Find All Solutions Within $$ (0, 2\pi) $$:**
   
   We need to find all integer values of $$ k $$ such that $$ x $$ falls within $$ (0, 2\pi) $$.

$$
   0 < \frac{\pi}{16} + \frac{k\pi}{4} < 2\pi
   $$
   
   Solving for $$ k $$:
   $$
   -\frac{\pi}{16} < \frac{k\pi}{4} < \frac{31\pi}{16}
   $$
   
   Dividing by $$ \frac{\pi}{4} $$:
   $$
   -\frac{1}{4} < k < \frac{31}{4}
   $$
   
   Since $$ k $$ must be an integer, the possible values are $$ k = 0, 1, 2, 3, 4, 5, 6, 7 $$.

4. **List All Solutions:**
   
   Substitute each $$ k $$ into the equation $$ x = \frac{\pi}{16} + \frac{k\pi}{4} $$:
   
   $$
   \begin{align*}
   k = 0 &: \quad x = \frac{\pi}{16} \
   k = 1 &: \quad x = \frac{5\pi}{16} \
   k = 2 &: \quad x = \frac{9\pi}{16} \
   k = 3 &: \quad x = \frac{13\pi}{16} \
   k = 4 &: \quad x = \frac{17\pi}{16} \
   k = 5 &: \quad x = \frac{21\pi}{16} \
   k = 6 &: \quad x = \frac{25\pi}{16} \
   k = 7 &: \quad x = \frac{29\pi}{16} \
   \end{align*}
   $$
   
All these values lie within the interval $$ (0, 2\pi) $$.

**Final Answer:**
All real numbers congruent to π⁄16, 5π⁄16, 9π⁄16, 13π⁄16, 17π⁄16, 21π⁄16, 25π⁄16, or 29π⁄16 modulo 2π. Thus, the solutions in (0, 2π) are

x = π/16, 5π/16, 9π/16, 13π/16, 17π/16, 21π/16, 25π/16, and 29π/16.
The solutions for $$ x $$ in the range $$ (0, 2\pi) $$ are: $$ x = \frac{\pi}{16}, \frac{5\pi}{16}, \frac{9\pi}{16}, \frac{13\pi}{16}, \frac{17\pi}{16}, \frac{21\pi}{16}, \frac{25\pi}{16}, \frac{29\pi}{16} $$

Simplify and solve the equation: for all solutions of in the range .

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