Equations involving tan(x) and special cases

The equation tan(x) = tan(a)

The tangent has a shorter period than cosine and sine: it is pi (not 2*pi). The rule is therefore simpler:

tan(x) = tan(a) <=> x = a + kπ, where k is a relative integer.

Example: Solve tan(x) = tan(π/4) on [0, 2π[. x = π/4 + kπ. For k = 0: x = π/4. For k = 1: x = π/4 + π = 5π/4. Solutions: x = π/4 or x = 5π/4.

Be aware of the domain of definition

tan(x) = sin(x)/cos(x) is not defined when cos(x) = 0, i.e. for x = pi/2 + k*pi. Before solving an equation involving tan, you must therefore always specify the domain of definition and exclude these values.

Special cases to remember

Equation	Solutions
cos(x) = 0	x = π/2 + kπ
cos(x) = 1	x = 2kπ
cos(x) = -1	x = π + 2kπ
sin(x) = 0	x = kπ
sin(x) = 1	x = π/2 + 2kπ
sin(x) = -1	x = -π/2 + 2kπ

Example: How many solutions does the equation sin(x) = 0 have on [0, 4π[? We have x = kπ, so x = 0, π, 2π, 3π: four solutions.

Common pitfall

Many pupils instinctively write “x = a + 2kπ” for the tangent, copying the rule for cosine or sine. This is incorrect: the period of tan is pi, not 2pi, so we miss half the solutions if we add 2kpi instead of k*pi.