Curriculum placement
Precalculus. This describes the skill, not a learner estimate.
Odd, even, and periodic trig functions
Explain odd even and periodic trigonometric relationships using the unit circle
Problem types
- Unit-circle symmetry
- Match a trigonometric value at a special angle with another expression
- Derive a symmetry from the unit circle
- Explain periodicity with the unit circle
- Which claims the unit circle justifies
Builds on
- Radians and the unit circle
- Exact values at special angles · Earlier idea
Once the values at special angles are known, reflecting or rotating the angle changes only the sign or the function, so other angles need no new values.
- Sine and cosine of complementary angles · Earlier idea
The cofunction relation sin(90 - x) = cos x is one of the symmetries of the sine and cosine on the unit circle.
Leads to
- Solving trigonometric equations · See it applied
Symmetry of the unit circle gives a second solution of a trigonometric equation in each period.
Related skills
- Polar and Cartesian coordinates · Related idea
Symmetry of sine and cosine shows in polar curves, such as symmetry across the axes.
- Period, amplitude, and midline · Related idea
Odd, even and periodic symmetry of sine and cosine appear on the graph as point symmetry, mirror symmetry and repetition every period.
- Symmetries of a polygon · Related idea
The symmetries of a regular polygon and of the unit circle explain why sine and cosine values repeat and reflect.
- The complex plane · Shared foundations
- Pythagorean and angle-sum identities · Shared foundations
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