Is the tangent function an odd function?

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Study for the NBCT Mathematics AYA Component 1 exam. Utilize flashcards and multiple-choice questions with detailed explanations for each question. Prepare efficiently for success in your teaching certification journey!

Multiple Choice

Is the tangent function an odd function?

Explanation:
Tangent has odd symmetry. A function is odd if f(-x) = -f(x) for all x in its domain. For tangent, tan(x) = sin x / cos x. Using the basic identities sin(-x) = -sin x and cos(-x) = cos x, we get tan(-x) = sin(-x)/cos(-x) = (-sin x)/cos x = -tan(x). This holds for all x where cos x ≠ 0 (i.e., x ≠ π/2 + kπ), so tan is an odd function on its domain.

Tangent has odd symmetry. A function is odd if f(-x) = -f(x) for all x in its domain. For tangent, tan(x) = sin x / cos x. Using the basic identities sin(-x) = -sin x and cos(-x) = cos x, we get tan(-x) = sin(-x)/cos(-x) = (-sin x)/cos x = -tan(x). This holds for all x where cos x ≠ 0 (i.e., x ≠ π/2 + kπ), so tan is an odd function on its domain.

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