Certainly! Let's solve the equation [tex]\( \tan(4x) - 1 = 0 \)[/tex] step-by-step.
1. Set the given equation equal to zero:
[tex]\[
\tan(4x) - 1 = 0
\][/tex]
2. Isolate the tangent function:
[tex]\[
\tan(4x) = 1
\][/tex]
3. Recall the angles where the tangent function equals 1:
The tangent function equals 1 at [tex]\( \frac{\pi}{4} + n\pi \)[/tex], where [tex]\( n \)[/tex] is any integer (since the tangent function has a period of [tex]\( \pi \)[/tex]).
4. Set [tex]\( 4x \)[/tex] equal to the general solution for the angle:
[tex]\[
4x = \frac{\pi}{4} + n\pi
\][/tex]
5. Solve for [tex]\( x \)[/tex] by dividing both sides by 4:
[tex]\[
x = \frac{\frac{\pi}{4} + n\pi}{4}
\][/tex]
6. Simplify the expression:
Break down the numerator and divide each term by 4:
[tex]\[
x = \frac{\pi}{4 \cdot 4} + \frac{n\pi}{4}
\][/tex]
[tex]\[
x = \frac{\pi}{16} + \frac{n\pi}{4}
\][/tex]
7. State the final solution:
[tex]\(
x = \frac{\pi}{16} + n\pi/4, \quad \text{where} \; n \; \text{is an integer}
\)[/tex]
Thus, the solutions to the equation [tex]\(\tan(4x) - 1 = 0\)[/tex] are:
[tex]\[
x = \frac{\pi}{16} + \frac{n\pi}{4}, \quad \text{where} \; n \; \text{is an integer}
\][/tex]
