AP Precalculus Section 3.15 Example: Describing Distances of the Polar Function r=2cos(𝜃)

Let's do so AP Precalculus Practice Problems! Please subscribe!    / nickperich   We are looking at the polar function where the radius, called r, is given as two times the cosine of the angle theta. In polar coordinates, each point is determined by two things: the distance from the origin (that’s r) and the angle from the positive x-axis (that’s theta). Here, the radius depends directly on the angle. When the angle is zero, cosine of zero is one, so the radius is two. That means the point is two units away from the origin along the positive x-axis. As the angle increases toward ninety degrees, the cosine gets smaller. At ninety degrees, the cosine is zero, so the radius is zero. That means the point is right at the origin. Between ninety degrees and one hundred eighty degrees, the cosine becomes negative. Multiplying by two gives a negative radius. In polar terms, a negative radius means we move in the exact opposite direction of the angle. So, the point is reflected across the origin. As the angle approaches one hundred eighty degrees, the radius again approaches negative two, putting the point two units away but in the opposite direction along the negative x-axis. Overall, this function produces a shape known as a circle, but it is described in polar terms, where the distance from the origin grows and shrinks depending on the angle. The farthest point from the origin is two units, and the closest is zero at certain angles. The key idea: the radius changes smoothly as the angle changes, following the pattern of the cosine function, which creates a circle that is offset from the origin along the x-axis. The Topics covered in AP Precalculus are... 1.1 Change in Tandem 1.2 Rates of Change 1.3 Rates of Change in Linear and Quadratic Functions 1.4 Polynomial Functions and Rates of Change 1.5 Polynomial Functions and Complex Zeros 1.6 Polynomial Functions and End Behavior 1.7 Rational Functions and End Behavior 1.8 Rational Functions and Zeros 1.9 Rational Functions and Vertical Asymptotes 1.10 Rational Functions and Holes 1.11 Equivalent Representations of Polynomial and Rational Expressions 1.12 Transformations of Functions 1.13 Function Model Selection and Assumption Articulation 1.14 Function Model Construction and Application 2.1 Change in Arithmetic and Geometric Sequences 2.2 Change in Linear and Exponential Functions 2.3 Exponential Functions 2.4 Exponential Function Manipulation 2.5 Exponential Function Context and Data Modeling 2.6 Competing Function Model Validation 2.7 Composition of Functions 2.8 Inverse Functions 2.9 Logarithmic Expressions 2.10 Inverses of Exponential Functions 2.11 Logarithmic Functions 2.12 Logarithmic Function Manipulation 2.13 Exponential and Logarithmic Equations and Inequalities 2.14 Logarithmic Function Context and Data Modeling 2.15 Semi-log Plots 3.1 Periodic Phenomena 3.2 Sine, Cosine, and Tangent 3.3 Sine and Cosine Function Values 3.4 Sine and Cosine Function Graphs 3.5 Sinusoidal Functions 3.6 Sinusoidal Function Transformations 3.7 Sinusoidal Function Context and Data Modeling 3.8 The Tangent Function 3.9 Inverse Trigonometric Functions 3.10 Trigonometric Equations and Inequalities 3.11 The Secant, Cosecant, and Cotangent Functions 3.12 Equivalent Representations of Trigonometric Functions 3.13 Trigonometry and Polar Coordinates 3.14 Polar Function Graphs 3.15 Rates of Change in Polar Functions I have many informative videos for Pre-Algebra, Algebra 1, Algebra 2, Geometry, Pre-Calculus, and Calculus. Please check it out: / nickperich Nick Perich Norristown Area High School Norristown Area School District Norristown, Pa #APPrecalculus #PreCalcProblems #MathMinds #AdvancedPreCalc #TrigTales #PrecalcPuzzles #FunctionFiesta #GraphGoals #CalcReady #PreCalcLife #AlgebraicAdventures #DerivativeDreams #IntegrationInsights #MathematicsMagic #PreCalcReview #PrecalcConcepts #LogarithmLove #TrigonometryTips #MathMastermind #APCalcPrep #Mathematics #MathMinds #Math #Maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study