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AP Precalculus Unit 3: Trigonometric & Polar Functions
On the Exam:
30%-35% of the Multiple Choice Section Score
2 of 4 Free Response Questions (Calculator Required)		Back to Scope and Sequence
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The Story of AP Precalculus, Unit 3 : Trigonometric & Polar Functions
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In Unit 3, students explore trigonometric functions and their relation to the angles and arcs of a circle. Since their output values repeat with every full revolution around the circle, trigonometric functions are ideal for modeling periodic, or repeated pattern phenomena, such as: the highs and lows of a wave, the blood pressure produced by a heart, and the angle from the North Pole to the Sun year to year. Furthermore, periodicity is found in human inventions and social phenomena. For example, moving parts of an analog clock are modeled by a trigonometric function with respect to each other or with respect to time; traffic flow at an intersection over the course of a week demonstrates daily periodicity; and demand for a particular product over the course of a year falls into an annually repeating pattern. Polar functions, which are also explored in this unit, have deep ties to trigonometric functions as they are both based on the circle. Polar functions are defined on the polar coordinate system that uses the circular concepts of radii and angles to describe location instead of rectangular concepts of left-right and up-down, which students have worked with previously. Trigonometry serves as the bridge between the two systems.
Unit 3 Learning Notes
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Building Mathematical Practices: Trigonometric & Polar Functions
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Students should have multiple experiences transitioning among, and communicating about, the various representations of trigonometric functions, especially sinusoidal functions. It is important that, in addition to solving trigonometric equations and finding equivalent trigonometric expressions, students build sinusoidal models with and without technology and practice constructing different representations. As students transition to thinking in the polar plane, they will refine their communications related to characteristics of functions. The more casual language that students may have adopted such as “goes up” and “goes down” will need to be replaced with more careful language that addresses a function’s behavior related to angles and radii.
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Preparing for the AP Exam: Trigonometric & Polar Functions
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As sinusoidal function content pairs well with each of the mathematical practices and skills in the course, students should practice applying each skill to each trigonometric function objective. On the AP Exam, students will need to describe characteristics of sinusoidal functions, such as amplitude, vertical shift, period, and phase shift, give reasons for why chosen values are consistent with information provided, calculate sinusoidal regressions using a graphing calculator, and apply trigonometric function models to contexts. Students should be practicing these skills in daily activities, so they are able to use them fluently on the AP Exam.
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Instructional Days
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Day 3.1.ADay 3.1.BDay 3.2.AOptional DayOptional Day
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I am learning to construct graphs of periodic relationships based on verbal representations. I am learning to describe key characteristics of a periodic function based on a verbal representation. I am learning to determine the sine, cosine, and tangent of an angle using the unit circle. The goal is to complete 3.2.A
in 1-3 days.

Success Criteria: See 3.2.A.
The goal is to complete 3.2.A
in 1-3 days.

Success Criteria: See 3.2.A.
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Day 3.3.AOptional DayOptional DayDay 3.4.AOptional Day
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I am learning to determine coordinates of points on a circle centered at the origin. The goal is to complete 3.3.A
in 1-3 days.

Success Criteria: See 3.3.A.
The goal is to complete 3.3.A
in 1-3 days.

Success Criteria: See 3.3.A.
I am learning to construct representations of the sine and cosine functions using the unit circle. The goal is to complete 3.4.A
in 1-3 days.

Success Criteria: See 3.4.A.
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Optional DayDay 3.5.AOptional DayOptional DayDay 3.6.A
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The goal is to complete 3.4.A
in 1-3 days.

Success Criteria: See 3.4.A.
I am learning to identify key characteristics of the sine and cosine functions. The goal is to complete 3.5.A
in 1-3 days.

Success Criteria: See 3.5.A.
The goal is to complete 3.5.A
in 1-3 days.

Success Criteria: See 3.5.A.
I am learning to identify the amplitude, vertical shift, period, and phase shift of a sinusoidal function.
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Optional DayOptional DayDay 3.7.AOptional DayOptional Day
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The goal is to complete 3.6.A
in 1-3 days.

Success Criteria: See 3.6.A.
The goal is to complete 3.6.A
in 1-3 days.

Success Criteria: See 3.6.A.
I am learning to construct sinusoidal function models of periodic phenomena by estimating key values. The goal is to complete 3.7.A
in 1-3 days.

Success Criteria: See 3.7.A.
The goal is to complete 3.7.A
in 1-3 days.

Success Criteria: See 3.7.A.
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Day 3.8.ADay 3.8.BDay 3.8.CDay 3.9.AOptional Day
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I am learning to construct representations of the tangent function using the unit circle. I am learning to describe key characteristics of the tangent function. I am learning to describe additive and multiplicative transformations involving the tangent function. I am learning to construct analytical and graphical representations of the inverse of the sine, cosine, and tangent functions over a restricted domain. The goal is to complete 3.9.A
in 1-3 days.

Success Criteria: See 3.9.A.
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Optional DayDay 3.10.AOptional DayOptional DayOptional Day
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The goal is to complete 3.9.A
in 1-3 days.

Success Criteria: See 3.9.A.
I am learning to solve equations and inequalities involving trigonometric functions. The goal is to complete 3.10.A
in 1-4 days.

Success Criteria: See 3.10.A.
The goal is to complete 3.10.A
in 1-4 days.

Success Criteria: See 3.10.A.
The goal is to complete 3.10.A
in 1-4 days.

Success Criteria: See 3.10.A.
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Day 3.11.AOptional DayDay 3.12.ADay 3.12.BDay 3.12.C
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I am learning to identify key characteristics of functions that involve quotients of the sine and cosine functions. The goal is to complete 3.11.A
in 1-2 days.

Success Criteria: See 3.11.A.
I am learning to construct sinusoidal function models of periodic phenomena by estimating key values. I am learning to rewrite trigonometric expressions in equivalent forms with sine and cosine sum identities. I am learning to solve equations using equivalent analytic representations of trigonometric functions.
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Optional DayDay 3.13.AOptional DayOptional DayDay 3.14.A
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The goal is to complete 3.12.A-3.12.C
in 3- 4 days.

Success Criteria: See 3.12.A, 3.12.B, and/or 3.12.C.
I am learning to determine the location of a point in the plane using both rectangular and polar coordinates. The goal is to complete 3.13.A
in 1-3 days.

Success Criteria: See 3.13.A.
The goal is to complete 3.13.A
in 1-3 days.

Success Criteria: See 3.13.A.
I am learning to construct graphs of polar functions.
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Optional DayOptional DayDay 3.15.AOptional DayOptional Day
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The goal is to complete 3.14.A
in 1-3 days.

Success Criteria: See 3.14.A.
The goal is to complete 3.14.A
in 1-3 days.

Success Criteria: See 3.14.A.
I am learning to describe characteristics of the graph of a polar function. The goal is to complete 3.15.A
in 1-3 days.

Success Criteria: See 3.15.A.
The goal is to complete 3.15.A
in 1-3 days.

Success Criteria: See 3.15.A.
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