Find the complete general solution to basic trig equations of the form sin(x) = k, cos(x) = k, or tan(x) = k over all real numbers. Solutions are expressed as infinite families using n ∈ ℤ, with exact π-fraction principal values and a unit circle diagram.
1/2, sqrt(3)/2, -1The trig functions sin and cos have a period of 2π — every complete revolution of the unit circle brings you back to the same point. Tan has a shorter period of π.
This means if x₀ solves sin(x) = k, then so does x₀ ± 2π, x₀ ± 4π, and so on for every integer multiple. We capture all of these with:
sin/cos: x = x₀ + 2πn, n ∈ ℤ
tan: x = x₀ + πn, n ∈ ℤ
The notation n ∈ ℤ means n can be any integer: …−2, −1, 0, 1, 2, … Each value of n gives a distinct solution.
For sin(x) = k: the sine function takes the same value at two angles per period — the principal value x₀ ∈ [−π/2, π/2] and its supplement π − x₀. This symmetry about the y-axis creates two distinct families:
x = x₀ + 2πn and x = π − x₀ + 2πn
For cos(x) = k: cos is symmetric about the x-axis — it takes the same value at ±x₀. So there are also two families: x = x₀ + 2πn and x = −x₀ + 2πn. (Exception: k = ±1 gives only one family.)
For tan(x) = k: tan is one-to-one within each half-period (−π/2, π/2), so each period has exactly one solution. This means one family with period π:
x = x₀ + πn
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