NY Next Generation Math AII-F.TF.4

MathGrades 9–12Extend the domain of trigonometric functions using the unit circle.

The Standard

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

New York State Next Generation Mathematics Learning Standards

Teacher's field guide

What Students Need to Do: Students use unit-circle coordinates to explain the symmetry and periodic behavior of sine, cosine, and tangent. They identify which functions are odd or even and justify repeated values.
What Mastery Looks Like: The student uses reflected or rotated unit-circle points to derive a symmetry relationship. The student states the correct period for sine, cosine, or tangent and supports it geometrically.
Common Misconceptions: Students may call every trigonometric function even or assign tangent the same period as sine. They may memorize identities without linking them to unit-circle coordinates.
How to Assess It: Ask students to use two related unit-circle angles to justify one odd or even identity and state the function's period.

Lesson moves

Fold a transparent unit circle across axes and compare coordinates at reflected angles.
Ask why tangent repeats after pi while sine and cosine repeat after two pi.
Play symmetry match with angle pairs, coordinate pairs, and sine, cosine, or tangent identities.
Connect repeating daylight or tide graphs to periodicity while separating the model from the geometric proof.

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