Find all solutions of the equation. Express the solutions in radians in the form a + nπ, where a is in [0, 2π).
tan x = 0
We are tasked with solving the trigonometric equation \(\tan x = 0\). The tangent function is defined as the ratio of sine to cosine: \(\tan x = \frac{\sin x}{\cos x}\). For the expression to be defined, we must have \(\cos x eq 0\). This means \(x eq \frac{\pi}{2} + k\pi\) for any integer \(k\). Given the definition, \(\tan x = 0\) is equivalent to \(\sin x = 0\) subject to the condition \(\cos x eq 0\).
We look for values of \(x\) such that \(\sin x = 0\). Within the interval \([0, 2\pi)\), the sine function is equal to zero at \(x = 0\) and \(x = \pi\). We verify the domain condition \(\cos x eq 0\) for these solutions:
Both solutions \(x = 0\) and \(x = \pi\) are valid.
The solutions we found in the range \([0, 2\pi)\) are \(0\) and \(\pi\). The question asks for the solutions in the form \(a + n\pi\), where \(a \in [0, 2\pi)\).
Notice that both sets describe the same set of points on the number line. Any integer multiple of \(\pi\) (i.e., \(\dots, -2\pi, -\pi, 0, \pi, 2\pi, \dots\)) satisfies \(\tan x = 0\). We can represent these solutions using \(a=0\) and \(n\) as any integer, yielding \(x = n\pi\).
The equation \(\tan x = 0\) often appears in wave mechanics and signal processing when determining the nodes of a standing wave. For instance, if a string is fixed at both ends, the points where the wave displacement is zero are related to the zeros of the sine or tangent functions. Understanding that \(\tan x = 0\) occurs at integer multiples of \(\pi\) helps in calculating the resonance frequencies and nodes of such physical systems.