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AP Precalculus Quiz

AP Precalculus Quiz: Trigonometry And Polar Coordinates

Practice Trigonometry And Polar Coordinates in AP Precalculus with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

Question 1 / 16

0 of 16 answered

A rescue helicopter is located at polar coordinates (r,θ)=(24, 210∘)(r,\theta)=(24,\,210^\circ)(r,θ)=(24,210∘) relative to a base, where rrr is kilometers. The base uses rectangular coordinates with xxx positive east and yyy positive north. The angle is given in degrees and measured counterclockwise from the positive xxx-axis. Convert using exact trigonometric values, not decimals. Convert the polar coordinates to rectangular coordinates (x,y)(x,y)(x,y) for the helicopter.​

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What this quiz covers

This quiz focuses on Trigonometry And Polar Coordinates, giving you a quick way to practice the rules, question types, and explanations that matter most for AP Precalculus.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

A rescue helicopter is located at polar coordinates (r,θ)=(24, 210∘)(r,\theta)=(24,\,210^\circ)(r,θ)=(24,210∘) relative to a base, where rrr is kilometers. The base uses rectangular coordinates with xxx positive east and yyy positive north. The angle is given in degrees and measured counterclockwise from the positive xxx-axis. Convert using exact trigonometric values, not decimals. Convert the polar coordinates to rectangular coordinates (x,y)(x,y)(x,y) for the helicopter.​

  1. (123, 12)\left(12\sqrt{3},\,12\right)(123​,12)
  2. (−123, −12)\left(-12\sqrt{3},\,-12\right)(−123​,−12) (correct answer)
  3. (−12, −123)\left(-12,\,-12\sqrt{3}\right)(−12,−123​)
  4. (12, −123)\left(12,\,-12\sqrt{3}\right)(12,−123​)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting polar coordinates to rectangular form with angles in degrees. The conversion formulas x = r cos(θ) and y = r sin(θ) require evaluating trigonometric functions at specific angles. In this scenario, the helicopter at (24, 210°) needs conversion, requiring evaluation of cos(210°) and sin(210°) using reference angles. Choice B is correct because it accurately applies the formulas: x = 24 cos(210°) = 24(-√3/2) = -12√3 and y = 24 sin(210°) = 24(-1/2) = -12, as 210° is in the third quadrant where both cosine and sine are negative. Choice C is incorrect because it reverses the roles of sine and cosine in the calculation, a common error when students confuse which function to use for x versus y. To help students: Draw the angle to identify the quadrant and use reference angles to find exact values. Remember that 210° = 180° + 30°, making it a third-quadrant angle with reference angle 30°.

Question 2

A rotating beacon’s intensity model uses the expression I(θ)=sin⁡θcos⁡θI(\theta)=\sin\theta\cos\thetaI(θ)=sinθcosθ, where θ\thetaθ is measured in radians. A technician rewrites the model using a single trigonometric function to simplify computations. Use exactly one standard identity and do not approximate. The angle variable remains in radians throughout. Using the identity, simplify the expression for I(θ)I(\theta)I(θ).​

  1. sin⁡(2θ)\sin(2\theta)sin(2θ)
  2. 12sin⁡(2θ)\tfrac{1}{2}\sin(2\theta)21​sin(2θ) (correct answer)
  3. 12cos⁡(2θ)\tfrac{1}{2}\cos(2\theta)21​cos(2θ)
  4. sin⁡2θ\sin^2\thetasin2θ

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on simplifying expressions using double-angle identities. The double-angle identity sin(2θ) = 2sin(θ)cos(θ) allows products of sine and cosine to be rewritten as a single function. In this scenario, the beacon intensity I(θ) = sin(θ)cos(θ) needs simplification using this identity, requiring rearrangement to solve for the product. Choice B is correct because it accurately applies the identity: since sin(2θ) = 2sin(θ)cos(θ), we have sin(θ)cos(θ) = (1/2)sin(2θ). Choice A is incorrect because it doubles the expression instead of halving it, showing confusion about which direction to apply the identity. To help students: Write the double-angle identity clearly and practice solving for different parts of the equation. Use substitution to verify your answer by checking specific angle values.

Question 3

A drone’s onboard map stores its displacement as rectangular coordinates (x,y)=(43, 4)(x,y)=(4\sqrt{3},\,4)(x,y)=(43​,4) meters from launch. The navigation module converts this to polar form (r,θ)(r,\theta)(r,θ) for steering. Let θ\thetaθ be measured in degrees from the positive xxx-axis, counterclockwise. Report an exact angle in degrees and an exact radius. Convert the rectangular coordinates to polar coordinates (r,θ)(r,\theta)(r,θ) for the drone.​

  1. (8, 30∘)(8,\,30^\circ)(8,30∘) (correct answer)
  2. (8, 60∘)(8,\,60^\circ)(8,60∘)
  3. (43, 30∘)(4\sqrt{3},\,30^\circ)(43​,30∘)
  4. (8, π/6)(8,\,\pi/6)(8,π/6)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting rectangular coordinates to polar form with angles in degrees. The conversion requires calculating r = √(x² + y²) and θ = arctan(y/x), then converting the angle to degrees. In this scenario, the drone at (4√3, 4) meters needs conversion to polar form, requiring careful calculation of both magnitude and angle. Choice A is correct because it accurately computes r = √(48 + 16) = √64 = 8 and θ = arctan(4/(4√3)) = arctan(1/√3) = 30°, as tan(30°) = 1/√3. Choice B is incorrect because 60° would give tan(60°) = √3, not 1/√3, showing confusion about which angle has which tangent value. To help students: Create a reference table of common angles and their trigonometric values. Practice recognizing ratios like 1/√3 and √3 to quickly identify the corresponding angles.

Question 4

A sound engineer studies a directional microphone pattern modeled by the polar equation r=4sin⁡θr=4\sin\thetar=4sinθ, with θ\thetaθ in radians. The engineer wants to know whether the pattern is symmetric about the line θ=π/2\theta=\pi/2θ=π/2. Use a standard symmetry test by substituting an equivalent angle expression. Do not use calculus or graphing technology. Analyze the graph of the polar equation and determine the symmetry about θ=π/2\theta=\pi/2θ=π/2.​

  1. Yes, symmetric about θ=π/2\theta=\pi/2θ=π/2. (correct answer)
  2. Yes, symmetric about the polar axis.
  3. No, only symmetric about the origin.
  4. No, it has no line symmetry.

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on analyzing symmetry in polar graphs. A polar curve is symmetric about the line θ = π/2 if replacing θ with π - θ yields an equivalent equation. In this scenario, the microphone pattern r = 4sin(θ) needs testing for symmetry about θ = π/2, requiring substitution and use of the identity sin(π - θ) = sin(θ). Choice A is correct because substituting θ with π - θ gives r = 4sin(π - θ) = 4sin(θ), which is the original equation, confirming symmetry about θ = π/2. Choice B is incorrect because symmetry about the polar axis (θ = 0) would require r(θ) = r(-θ), but sin(-θ) = -sin(θ), not sin(θ). To help students: Learn the three main symmetry tests for polar curves - about the polar axis, about θ = π/2, and about the origin. Practice applying trigonometric identities to verify each type of symmetry.

Question 5

A hiker walks 999 km at a bearing of 45∘45^\circ45∘ south of east, corresponding to θ=−π4\theta=-\tfrac{\pi}{4}θ=−4π​ rad. Convert (r,θ)=(9, −π4)(r,\theta)=(9,\,-\tfrac{\pi}{4})(r,θ)=(9,−4π​) to rectangular coordinates (x,y)(x,y)(x,y).

  1. (922, 922)(\tfrac{9\sqrt{2}}{2},\,\tfrac{9\sqrt{2}}{2})(292​​,292​​)
  2. (922, −922)(\tfrac{9\sqrt{2}}{2},\,-\tfrac{9\sqrt{2}}{2})(292​​,−292​​) (correct answer)
  3. (−922, −922)(-\tfrac{9\sqrt{2}}{2},\,-\tfrac{9\sqrt{2}}{2})(−292​​,−292​​)
  4. (−922, 922)(-\tfrac{9\sqrt{2}}{2},\,\tfrac{9\sqrt{2}}{2})(−292​​,292​​)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting polar coordinates with negative angles to rectangular coordinates. Negative angles are measured clockwise from the positive x-axis, and bearings must be properly interpreted in standard position. In this scenario, the hiker's bearing of 45° south of east corresponds to -π/4 radians, placing the direction in the fourth quadrant. Choice B is correct because x = 9cos(-π/4) = 9(√2/2) = 9√2/2 and y = 9sin(-π/4) = 9(-√2/2) = -9√2/2, correctly showing positive x and negative y for the fourth quadrant. Choice A is incorrect because it shows both coordinates as positive, which would place the point in the first quadrant rather than the fourth quadrant as required by the negative angle. To help students: Visualize negative angles as clockwise rotations and practice converting bearings to standard position angles. Remember that cos(-θ) = cos(θ) but sin(-θ) = -sin(θ).

Question 6

A circular sprinkler sprays a point located at (r,θ)=(10, 210∘)(r,\theta)=(10,\,210^\circ)(r,θ)=(10,210∘) from the valve. Convert these polar coordinates to rectangular coordinates (x,y)(x,y)(x,y) in meters.

  1. (5, −53)(5,\,-5\sqrt{3})(5,−53​)
  2. (−5, 53)(-5,\,5\sqrt{3})(−5,53​)
  3. (−53, −5)(-5\sqrt{3},\,-5)(−53​,−5) (correct answer)
  4. (−53, 5)(-5\sqrt{3},\,5)(−53​,5)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting polar coordinates with degree angle measures to rectangular coordinates. The conversion requires evaluating trigonometric functions at 210°, which is in the third quadrant where both cosine and sine are negative. In this scenario, the sprinkler point at (10, 210°) requires calculating x = 10cos(210°) and y = 10sin(210°). Choice C is correct because cos(210°) = -cos(30°) = -√3/2 and sin(210°) = -sin(30°) = -1/2, giving x = 10(-√3/2) = -5√3 and y = 10(-1/2) = -5. Choice D is incorrect because it has the wrong sign for y, placing the point in the second quadrant instead of the third quadrant. To help students: Use reference angles to find trigonometric values, remembering that 210° = 180° + 30°. Always verify that the final coordinates match the expected quadrant based on the original angle.

Question 7

A robotic arm’s vibration analysis produces the expression E(θ)=1−cos⁡(2θ)2E(\theta)=\dfrac{1-\cos(2\theta)}{2}E(θ)=21−cos(2θ)​, with θ\thetaθ in radians. The control system prefers a simpler equivalent form using a squared sine function. Use a single trigonometric identity and keep the expression exact. No degree measures appear in this model. Using the identity, simplify the expression for E(θ)E(\theta)E(θ).​

  1. cos⁡2θ\cos^2\thetacos2θ
  2. sin⁡2θ\sin^2\thetasin2θ (correct answer)
  3. 12sin⁡(2θ)\tfrac{1}{2}\sin(2\theta)21​sin(2θ)
  4. 1−sin⁡2θ1-\sin^2\theta1−sin2θ

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on simplifying expressions using the cosine double-angle identity. The identity cos(2θ) = 1 - 2sin²(θ) can be rearranged to express complex forms in terms of squared trigonometric functions. In this scenario, E(θ) = (1 - cos(2θ))/2 needs simplification, requiring substitution of the double-angle identity and algebraic manipulation. Choice B is correct because it accurately applies the identity: substituting cos(2θ) = 1 - 2sin²(θ) gives (1 - (1 - 2sin²(θ)))/2 = 2sin²(θ)/2 = sin²(θ). Choice D is incorrect because 1 - sin²(θ) = cos²(θ), showing confusion between complementary squared functions. To help students: Learn multiple forms of the cosine double-angle identity and practice choosing the most useful form for each problem. Verify answers by substituting specific angle values to check equivalence.

Question 8

An engineer locates a bolt at (x,y)=(4, −4)(x,y)=(4,\,-4)(x,y)=(4,−4) centimeters from the origin. Convert to polar form (r,θ)(r,\theta)(r,θ) with θ\thetaθ in radians, 0≤θ<2π0\le\theta<2\pi0≤θ<2π.

  1. (42, π4)(4\sqrt{2},\,\tfrac{\pi}{4})(42​,4π​)
  2. (42, 7π4)(4\sqrt{2},\,\tfrac{7\pi}{4})(42​,47π​) (correct answer)
  3. (8, 7π4)(8,\,\tfrac{7\pi}{4})(8,47π​)
  4. (42, 5π4)(4\sqrt{2},\,\tfrac{5\pi}{4})(42​,45π​)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting rectangular coordinates to polar form with angle restrictions. Polar coordinates require calculating r = √(x² + y²) and determining θ based on the quadrant and the constraint 0 ≤ θ < 2π. In this scenario, the point (4, -4) is in the fourth quadrant, requiring careful angle calculation. Choice B is correct because it accurately computes r = √(4² + (-4)²) = √32 = 4√2, and since the point is in the fourth quadrant with equal magnitude coordinates, θ = 2π - π/4 = 7π/4. Choice A is incorrect because π/4 would place the point in the first quadrant where both coordinates are positive, not in the fourth quadrant where y is negative. To help students: Draw the point on a coordinate plane to visualize its quadrant location. Remember that in the fourth quadrant, angles are measured as 2π minus the reference angle, ensuring 0 ≤ θ < 2π.

Question 9

A rotating radar completes 150∘150^\circ150∘ of sweep to track a storm cell. Determine the equivalent angle measure in radians.

  1. 5π12\tfrac{5\pi}{12}125π​
  2. 5π6\tfrac{5\pi}{6}65π​ (correct answer)
  3. 3π5\tfrac{3\pi}{5}53π​
  4. 7π6\tfrac{7\pi}{6}67π​

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting angle measurements from degrees to radians. The conversion requires multiplying degrees by π/180° to obtain the radian measure. In this scenario, the radar completes 150° of sweep, requiring conversion to radians using the standard conversion factor. Choice B is correct because 150° × (π/180°) = 150π/180 = 5π/6 radians, properly simplifying the fraction. Choice A is incorrect because 5π/12 would correspond to 75°, suggesting the student may have divided 150° by 2 before converting or made an arithmetic error in simplification. To help students: Set up the conversion as a fraction multiplication problem and cancel common factors systematically. Memorize common angle conversions like 30° = π/6, 60° = π/3, and 90° = π/2 to check reasonableness of answers.

Question 10

A signal’s phase shift uses the expression sin⁡θcos⁡θ\sin\theta\cos\thetasinθcosθ. Using the identity, simplify the expression for the phase shift in terms of sin⁡(2θ)\sin(2\theta)sin(2θ).

  1. sin⁡(2θ)\sin(2\theta)sin(2θ)
  2. 12sin⁡(2θ)\tfrac{1}{2}\sin(2\theta)21​sin(2θ) (correct answer)
  3. 12cos⁡(2θ)\tfrac{1}{2}\cos(2\theta)21​cos(2θ)
  4. 2sin⁡(2θ)2\sin(2\theta)2sin(2θ)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on applying the double angle identity for sine. The identity sin(2θ) = 2sin(θ)cos(θ) allows simplification of products of sine and cosine into a single trigonometric function. In this scenario, the expression sin(θ)cos(θ) needs to be rewritten using the double angle formula. Choice B is correct because rearranging the identity sin(2θ) = 2sin(θ)cos(θ) gives sin(θ)cos(θ) = (1/2)sin(2θ), properly accounting for the factor of 2. Choice A is incorrect because it omits the factor of 1/2, suggesting the student directly substituted without properly rearranging the identity. To help students: Write out the double angle identities clearly and practice algebraic manipulation to isolate different forms. Emphasize that sin(θ)cos(θ) is half of sin(2θ), not equal to it.

Question 11

A wave model includes 1−2sin⁡2θ1-2\sin^2\theta1−2sin2θ for an angle θ\thetaθ measured in radians. Using the identity, simplify this expression in terms of cos⁡(2θ)\cos(2\theta)cos(2θ).

  1. cos⁡(2θ)\cos(2\theta)cos(2θ) (correct answer)
  2. −cos⁡(2θ)-\cos(2\theta)−cos(2θ)
  3. sin⁡(2θ)\sin(2\theta)sin(2θ)
  4. 1−cos⁡(2θ)1-\cos(2\theta)1−cos(2θ)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on applying the cosine double angle identity. The identity cos(2θ) = 1 - 2sin²(θ) directly relates to the given expression and allows for simplification. In this scenario, the wave model expression 1 - 2sin²(θ) matches exactly with one form of the cosine double angle identity. Choice A is correct because the identity cos(2θ) = 1 - 2sin²(θ) shows that 1 - 2sin²(θ) = cos(2θ) directly, without any sign changes or additional terms. Choice B is incorrect because it includes a negative sign, suggesting confusion with the identity cos(2θ) = 2cos²(θ) - 1 or misremembering the formula. To help students: Memorize all three forms of the cosine double angle identity: cos(2θ) = cos²(θ) - sin²(θ) = 2cos²(θ) - 1 = 1 - 2sin²(θ). Practice recognizing which form matches the given expression structure.

Question 12

A lighthouse beacon is modeled by P(θ)=sin⁡θcsc⁡θP(\theta)=\dfrac{\sin\theta}{\csc\theta}P(θ)=cscθsinθ​, where θ\thetaθ is measured in radians. The maintenance log asks for a simplified expression to reduce computational error. Use exactly one reciprocal identity and keep the result in terms of sin⁡θ\sin\thetasinθ only. Do not introduce additional trigonometric functions. Using the identity, simplify the expression for P(θ)P(\theta)P(θ).​

  1. 111
  2. sin⁡2θ\sin^2\thetasin2θ (correct answer)
  3. csc⁡2θ\csc^2\thetacsc2θ
  4. sin⁡θ\sin\thetasinθ

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on simplifying expressions using reciprocal identities. The reciprocal identity csc(θ) = 1/sin(θ) allows complex fractions to be simplified by converting between reciprocal pairs. In this scenario, P(θ) = sin(θ)/csc(θ) needs simplification using the cosecant reciprocal identity. Choice B is correct because it accurately applies the identity: P(θ) = sin(θ)/csc(θ) = sin(θ)/(1/sin(θ)) = sin(θ) × sin(θ) = sin²(θ). Choice A is incorrect because it would result from incorrectly canceling sin(θ) with csc(θ) as if they were reciprocals that multiply to 1, rather than understanding that division by a reciprocal means multiplication. To help students: Remember that dividing by a fraction means multiplying by its reciprocal. Practice rewriting all six trigonometric functions in terms of sine and cosine to build fluency with identities.

Question 13

A boat travels 121212 km at a bearing of 30∘30^\circ30∘ east of north. Convert (r,θ)=(12, 60∘)(r,\theta)=(12,\,60^\circ)(r,θ)=(12,60∘) to rectangular coordinates (x,y)(x,y)(x,y) in kilometers.

  1. (63, 6)(6\sqrt{3},\,6)(63​,6)
  2. (6, 63)(6,\,6\sqrt{3})(6,63​) (correct answer)
  3. (−63, 6)(-6\sqrt{3},\,6)(−63​,6)
  4. (12cos⁡30∘, 12sin⁡30∘)(12\cos 30^\circ,\,12\sin 30^\circ)(12cos30∘,12sin30∘)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting polar coordinates to rectangular coordinates. Polar coordinates provide a system for describing locations in a plane using radius and angle, where rectangular coordinates use x = r cos(θ) and y = r sin(θ). In this scenario, a boat travels 12 km at a bearing of 30° east of north, which corresponds to θ = 60° from the positive x-axis (since bearings are measured from north clockwise, and standard angles are measured counterclockwise from east). Choice B is correct because it accurately applies the conversion formulas: x = 12 cos(60°) = 12(1/2) = 6 and y = 12 sin(60°) = 12(√3/2) = 6√3. Choice A is incorrect because it reverses the x and y values, a common error when students confuse which trigonometric function to use with each coordinate. To help students: Encourage drawing diagrams to visualize the angle measurement system and practice converting between bearings and standard position angles. Emphasize memorizing the conversion formulas x = r cos(θ) and y = r sin(θ) and the special angle values.

Question 14

A drone’s displacement is modeled by (r,θ)=(8, 5π6 rad)(r,\theta)=(8,\,\tfrac{5\pi}{6}\text{ rad})(r,θ)=(8,65π​ rad) from its launch point. Convert these polar coordinates to rectangular coordinates (x,y)(x,y)(x,y) in meters.

  1. (−43, 4)(-4\sqrt{3},\,4)(−43​,4) (correct answer)
  2. (43, 4)(4\sqrt{3},\,4)(43​,4)
  3. (−4, 43)(-4,\,4\sqrt{3})(−4,43​)
  4. (−43, −4)(-4\sqrt{3},\,-4)(−43​,−4)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting polar coordinates with radian angle measures to rectangular coordinates. Trigonometric functions allow the modeling of periodic phenomena, and the conversion requires evaluating cosine and sine at the given angle. In this scenario, the drone's displacement is given as (r,θ) = (8, 5π/6 rad), requiring calculation of x = r cos(θ) and y = r sin(θ) at this second quadrant angle. Choice A is correct because it accurately applies the conversion: x = 8 cos(5π/6) = 8(-√3/2) = -4√3 and y = 8 sin(5π/6) = 8(1/2) = 4, correctly placing the point in the second quadrant. Choice C is incorrect because it confuses the cosine and sine values, resulting in switched magnitudes for x and y coordinates. To help students: Use the unit circle to visualize angles in radians and their corresponding cosine and sine values. Practice identifying which quadrant an angle falls in and the signs of trigonometric functions in each quadrant.

Question 15

A lighthouse beam rotates at 12∘12^\circ12∘ per second for 151515 seconds. Determine the total angle swept, in radians, during this interval.

  1. π\piπ (correct answer)
  2. π2\tfrac{\pi}{2}2π​
  3. 2π2\pi2π
  4. 3π4\tfrac{3\pi}{4}43π​

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting degrees to radians for angular measurements. The conversion factor π radians = 180° is essential for translating between these two angle measurement systems. In this scenario, the lighthouse beam rotates at 12° per second for 15 seconds, giving a total rotation of 12° × 15 = 180°. Choice A is correct because 180° converts to radians as 180° × (π/180°) = π radians, representing a half rotation. Choice C is incorrect because 2π radians equals 360°, which would represent a full rotation rather than the half rotation that actually occurred. To help students: Practice dimensional analysis with the conversion factor π/180° to convert degrees to radians. Visualize common angles like 180° = π as half a circle and 360° = 2π as a full circle to build intuition.

Question 16

A surveyor records a point at (x,y)=(−3, 33)(x,y)=(-3,\,3\sqrt{3})(x,y)=(−3,33​) meters relative to a marker. Convert this location to polar form (r,θ)(r,\theta)(r,θ) with θ\thetaθ in radians.

  1. (6, 2π3)(6,\,\tfrac{2\pi}{3})(6,32π​) (correct answer)
  2. (6, π3)(6,\,\tfrac{\pi}{3})(6,3π​)
  3. (33, 2π3)(3\sqrt{3},\,\tfrac{2\pi}{3})(33​,32π​)
  4. (6, −2π3)(6,\,-\tfrac{2\pi}{3})(6,−32π​)

Explanation: This question tests AP Precalculus skills in trigonometric and polar functions, specifically focusing on converting rectangular coordinates to polar form. The conversion requires finding both the radius r = √(x² + y²) and the angle θ = arctan(y/x), with careful attention to quadrant. In this scenario, the point (-3, 3√3) is in the second quadrant, requiring proper angle determination. Choice A is correct because it accurately calculates r = √((-3)² + (3√3)²) = √(9 + 27) = √36 = 6, and θ = arctan(3√3/(-3)) = arctan(-√3), which gives 2π/3 in the second quadrant (not -π/3 which would be fourth quadrant). Choice B is incorrect because it uses π/3, which would place the point in the first quadrant instead of the second quadrant where x is negative. To help students: Emphasize checking which quadrant the original point is in and adjusting the angle accordingly. Practice using reference angles and adding π when the point is in the second or third quadrant.