Unit 3
The unit that introduces periodicity. It builds the unit circle, turns it into sinusoidal functions with four adjustable parameters, uses those to model repeating phenomena, and then changes coordinate systems entirely to look at polar functions.
Lesson 3.1
A periodic function repeats after a fixed positive input shift. The smallest positive repeat time, when it exists, is the fundamental period. A repeated output value alone is insufficient to establish a full repeat of the function.
Lesson 3.2
A point on the unit circle at angle theta has coordinates cosine theta and sine theta. Reference angles give magnitudes, while the quadrant sets signs. Angles that differ by full revolutions reach the same point.
Lesson 3.3
In a transformed sinusoid, the absolute outside coefficient gives amplitude, the vertical shift gives the midline, and the absolute inside frequency determines period. These interpretations use a sine or cosine model with a nonzero frequency.
Lesson 3.4
Factor the coefficient of x before reading a horizontal shift. A positive h in the form x minus h shifts the graph right. The order matters: reading the constant before factoring usually gives the wrong shift.
Lesson 3.5
Choose a sine or cosine model from the range, repeat time, and starting position. The coefficient of time uses radians per time unit. Include the physical domain and check whether the initial value and direction agree with the context.
Lesson 3.6
Use a reference angle and all compatible quadrants, then restrict to the stated interval. Factoring is safer than dividing by a trigonometric expression that might be zero. A general solution includes periodic repeats.
Lesson 3.7
In a polar function, radius changes with angle. Increasing radius means motion outward only when the radius is nonnegative; a negative radius places a point in the opposite direction. Evaluate signed radii and convert selected points before describing the curve.
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