Trigonometry and Polar Coordinates
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AP Precalculus › Trigonometry and Polar Coordinates
A rescue helicopter is located at polar coordinates $(r,\theta)=(24,,210^\circ)$ relative to a base, where $r$ is kilometers. The base uses rectangular coordinates with $x$ positive east and $y$ positive north. The angle is given in degrees and measured counterclockwise from the positive $x$-axis. Convert using exact trigonometric values, not decimals. Convert the polar coordinates to rectangular coordinates $(x,y)$ for the helicopter.
$\left(-12,,-12\sqrt{3}\right)$
$\left(-12\sqrt{3},,-12\right)$
$\left(12\sqrt{3},,12\right)$
$\left(12,,-12\sqrt{3}\right)$
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°.
A rotating beacon’s intensity model uses the expression $I(\theta)=\sin\theta\cos\theta$, where $\theta$ 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 $I(\theta)$.
$\tfrac{1}{2}\cos(2\theta)$
$\sin^2\theta$
$\sin(2\theta)$
$\tfrac{1}{2}\sin(2\theta)$
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.
A drone’s onboard map stores its displacement as rectangular coordinates $(x,y)=(4\sqrt{3},,4)$ meters from launch. The navigation module converts this to polar form $(r,\theta)$ for steering. Let $\theta$ be measured in degrees from the positive $x$-axis, counterclockwise. Report an exact angle in degrees and an exact radius. Convert the rectangular coordinates to polar coordinates $(r,\theta)$ for the drone.
$(4\sqrt{3},,30^\circ)$
$(8,,\pi/6)$
$(8,,30^\circ)$
$(8,,60^\circ)$
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.
A hiker walks $9$ km at a bearing of $45^\circ$ south of east, corresponding to $\theta=-\tfrac{\pi}{4}$ rad. Convert $(r,\theta)=(9,,-\tfrac{\pi}{4})$ to rectangular coordinates $(x,y)$.
$(-\tfrac{9\sqrt{2}}{2},,-\tfrac{9\sqrt{2}}{2})$
$(\tfrac{9\sqrt{2}}{2},,-\tfrac{9\sqrt{2}}{2})$
$(-\tfrac{9\sqrt{2}}{2},,\tfrac{9\sqrt{2}}{2})$
$(\tfrac{9\sqrt{2}}{2},,\tfrac{9\sqrt{2}}{2})$
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).
A circular sprinkler sprays a point located at $(r,\theta)=(10,,210^\circ)$ from the valve. Convert these polar coordinates to rectangular coordinates $(x,y)$ in meters.
$(5,,-5\sqrt{3})$
$(-5\sqrt{3},,5)$
$(-5,,5\sqrt{3})$
$(-5\sqrt{3},,-5)$
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.
A robotic arm’s vibration analysis produces the expression $E(\theta)=\dfrac{1-\cos(2\theta)}{2}$, with $\theta$ 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 $E(\theta)$.
$1-\sin^2\theta$
$\cos^2\theta$
$\sin^2\theta$
$\tfrac{1}{2}\sin(2\theta)$
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.
An engineer locates a bolt at $(x,y)=(4,,-4)$ centimeters from the origin. Convert to polar form $(r,\theta)$ with $\theta$ in radians, $0\le\theta<2\pi$.
$(4\sqrt{2},,\tfrac{5\pi}{4})$
$(8,,\tfrac{7\pi}{4})$
$(4\sqrt{2},,\tfrac{\pi}{4})$
$(4\sqrt{2},,\tfrac{7\pi}{4})$
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.
A rotating radar completes $150^\circ$ of sweep to track a storm cell. Determine the equivalent angle measure in radians.
$\tfrac{5\pi}{12}$
$\tfrac{3\pi}{5}$
$\tfrac{5\pi}{6}$
$\tfrac{7\pi}{6}$
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.
A signal’s phase shift uses the expression $\sin\theta\cos\theta$. Using the identity, simplify the expression for the phase shift in terms of $\sin(2\theta)$.
$\tfrac{1}{2}\sin(2\theta)$
$2\sin(2\theta)$
$\tfrac{1}{2}\cos(2\theta)$
$\sin(2\theta)$
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.
A wave model includes $1-2\sin^2\theta$ for an angle $\theta$ measured in radians. Using the identity, simplify this expression in terms of $\cos(2\theta)$.
$1-\cos(2\theta)$
$-\cos(2\theta)$
$\cos(2\theta)$
$\sin(2\theta)$
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.