Describe the range of tan x.

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Multiple Choice

Describe the range of tan x.

Explanation:
Tangent represents a slope-like ratio sin x over cos x, and it can grow without bound in either direction as x approaches the vertical asymptotes where cos x is zero. Between those asymptotes, tan x sweeps from negative to positive infinity, crossing through zero at x = kπ. For any real number y, you can find an angle whose tangent equals y by taking x = arctan(y) plus any multiple of π. That shows tan x attains every real value, so the range is all real numbers. (The function isn’t defined where cos x = 0, i.e., at x = π/2 + kπ, but that doesn’t limit the range since tan x takes every real value elsewhere.)

Tangent represents a slope-like ratio sin x over cos x, and it can grow without bound in either direction as x approaches the vertical asymptotes where cos x is zero. Between those asymptotes, tan x sweeps from negative to positive infinity, crossing through zero at x = kπ. For any real number y, you can find an angle whose tangent equals y by taking x = arctan(y) plus any multiple of π. That shows tan x attains every real value, so the range is all real numbers. (The function isn’t defined where cos x = 0, i.e., at x = π/2 + kπ, but that doesn’t limit the range since tan x takes every real value elsewhere.)

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