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AP Precalculus Practice Test Unit 3 Question 43 Polar Graphs of rasintheta and racostheta
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Please subscribe! / nickperich • My AP Precalculus Practice Tests are carefully designed to help students build confidence for in-class assessments, support their work on AP Classroom assignments, and thoroughly prepare them for the AP Precalculus exam in May. • *AP Precalculus Practice Test: Unit 3, Question #43* asks about the polar graphs of the equations \\( r = a \\sin(\\theta) \\) and \\( r = a \\cos(\\theta) \\). • --- • **Key Concepts**: • 1. **Polar Equations**: • In polar coordinates, the equation \\( r = f(\\theta) \\) represents a curve where \\( r \\) is the radial distance from the origin and \\( \\theta \\) is the angle. • The graphs of \\( r = a \\sin(\\theta) \\) and \\( r = a \\cos(\\theta) \\) describe specific curves, and these depend on the value of \\( a \\). • 2. **Graph of \\( r = a \\sin(\\theta) \\)**: • This equation represents a **circle**. • The center of the circle is at \\( (0, \\frac{a}{2}) \\) in polar coordinates, and its radius is \\( \\frac{a}{2} \\). • When \\( a \\) is positive, the circle lies in the upper half of the plane. When \\( a \\) is negative, the circle lies in the lower half of the plane. • The graph forms a complete circle as \\( \\theta \\) ranges from 0 to \\( 2\\pi \\). • 3. **Graph of \\( r = a \\cos(\\theta) \\)**: • This also represents a **circle**. • The center of this circle is at \\( \\left( \\frac{a}{2}, 0 \ ight) \\) in polar coordinates, and its radius is \\( \\frac{a}{2} \\). • When \\( a \\) is positive, the circle is centered to the right of the origin. When \\( a \\) is negative, the circle is centered to the left of the origin. • The graph also completes a full rotation as \\( \\theta \\) goes from 0 to \\( 2\\pi \\). • --- • **Steps to Sketch the Graphs**: • 1. **Graph of \\( r = a \\sin(\\theta) \\)**: • When \\( a \\) is positive, the circle is centered at \\( (0, \\frac{a}{2}) \\) with a radius of \\( \\frac{a}{2} \\). • When \\( a \\) is negative, the circle is reflected over the x-axis but is still centered at \\( (0, \\frac{a}{2}) \\). • 2. **Graph of \\( r = a \\cos(\\theta) \\)**: • When \\( a \\) is positive, the circle is centered at \\( \\left( \\frac{a}{2}, 0 \ ight) \\) with a radius of \\( \\frac{a}{2} \\). • When \\( a \\) is negative, the circle is reflected over the y-axis but is still centered at \\( \\left( \\frac{a}{2}, 0 \ ight) \\). • --- • **Summary**: • The graph of \\( r = a \\sin(\\theta) \\) is a circle centered at \\( (0, \\frac{a}{2}) \\) with a radius of \\( \\frac{a}{2} \\), and the graph of \\( r = a \\cos(\\theta) \\) is a circle centered at \\( \\left( \\frac{a}{2}, 0 \ ight) \\) with a radius of \\( \\frac{a}{2} \\). • The placement and direction of the circle depend on whether \\( a \\) is positive or negative. • • 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 • #math #algebra #algebra2 #maths #math #shorts #funny #help #onlineclasses #onlinelearning #online #study
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