Back to Explain symmetry and periodicity — Problem 3 · Task Set 69

HSF.TF.A.4 Tutor Intake — Symmetry and Periodicity of Trig Functions

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Grade 9·11 problems·~14 min·Common Core Math - HS Functions·group·hsf-tf-a-4

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Concepts

Which statement correctly classifies tangent and provides a valid algebraic justification?

Tangent is even because $\tan(-x) = \dfrac{\cos(-x)}{\sin(-x)} = \dfrac{\cos x}{-\sin x} = -\tan(x)$ ,
so $\tan(-x) = -\tan(x)$ — confirming even symmetry.

Tangent is even because both sine and cosine are defined for all angles.

Tangent is odd because $\tan(-x) = \dfrac{\sin(-x)}{\cos(-x)} = \dfrac{-\sin x}{\cos x} = -\tan(x)$ .

Tangent is neither even nor odd because it has vertical asymptotes.