Sine and Cosine Function Values
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AP Precalculus › Sine and Cosine Function Values
A sound wave’s displacement is modeled by $y=\sin(\theta)$, where $\theta$ is the phase angle on the unit circle. At a particular time, the phase is $\theta=90^\circ=\pi/2$, measured counterclockwise from the positive $x$-axis. The unit circle has radius 1, so sine equals the $y$-coordinate. Use an exact value consistent with standard special angles. What is the sine of an angle measuring $\pi/2$ radians on the unit circle?
$1$
$0$
$-1$
$1/2$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Sine values represent the y-coordinate on the unit circle where the angle intersects. In this problem, the angle θ = π/2 (90°) is given for a sound wave model, requiring students to find its sine value. Choice B (1) is correct because at π/2 radians, the point on the unit circle is at (0, 1), making sin(π/2) = 1. Choice A (0) is incorrect as this is the cosine value at π/2, showing confusion between x and y coordinates at this quadrantal angle. Students should memorize the four quadrantal angles (0°, 90°, 180°, 270°) and their exact trigonometric values. Visualizing the unit circle helps students see that π/2 points straight up, giving maximum sine value.
An AC circuit has a phase angle $\theta=45^\circ=\pi/4$ between voltage and current, modeled on the unit circle. The cosine of $\theta$ represents the in-phase component magnitude for a 1-unit signal. Angles are measured in radians for computation, but the setup also states degrees. Use standard unit-circle exact values, not rounded approximations. Using the unit circle, what is the cosine value for $\pi/4$?
$\sqrt{2}/2$
$0.707$
$-\sqrt{2}/2$
$\sqrt{3}/2$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Cosine values represent the x-coordinate on the unit circle where the angle intersects. In this problem, the angle θ = π/4 (45°) is given in an AC circuit context, requiring students to find its cosine value. Choice A (√2/2) is correct because at π/4 radians, the point on the unit circle has coordinates (√2/2, √2/2), making cos(π/4) = √2/2. Choice D (0.707) is incorrect as it's a decimal approximation when the problem specifically asks for exact values, showing students must distinguish between exact and approximate forms. Students should memorize that at 45°, both sine and cosine equal √2/2. Using the isosceles right triangle with hypotenuse 1 helps derive this value exactly.
A sound wave model uses $y=\cos(\theta)$, where $\theta$ is a phase angle on the unit circle. At a specific instant, $\theta=180^\circ=\pi$, placing the point on the negative $x$-axis. The cosine equals the $x$-coordinate, so it can be read exactly from the unit circle. Use exact values and standard angle positions within $0\le\theta\le2\pi$. Using the unit circle, what is the cosine value for $\pi$?
$-1$
$0$
$-0$
$1$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Cosine values represent the x-coordinate on the unit circle where the angle intersects. In this problem, the angle θ = π (180°) is given for a sound wave model, requiring students to find its cosine value at this quadrantal angle. Choice C (-1) is correct because at π radians, the point on the unit circle is at (-1, 0), making cos(π) = -1. Choice B (0) is incorrect as this is the sine value at π, showing confusion between x and y coordinates at quadrantal angles. Students should memorize that π radians points directly left on the unit circle, giving the most negative cosine value. Understanding quadrantal angles as multiples of π/2 helps students quickly identify their exact trigonometric values.
An AC generator is analyzed with a phase angle $\theta$ on the unit circle, linking cosine to the real component. The measured value is $\cos\theta=-3/5$, and the phase lies in Quadrant II, consistent with angles like $150^\circ=5\pi/6$. Use $\sin^2\theta+\cos^2\theta=1$ to compute sine exactly, then apply the quadrant sign rule. Degrees and radians are both referenced, but the identity is purely algebraic. Find $\sin(\theta)$ given $\cos(\theta)$ and $\theta$ is in quadrant II.
$3/5$
$5/4$
$4/5$
$-4/5$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. The Pythagorean identity sin²θ + cos²θ = 1 connects sine and cosine values on the unit circle. In this problem, cos(θ) = -3/5 with θ in Quadrant II, requiring students to find sin(θ) using the identity. Choice A (4/5) is correct because sin²θ = 1 - cos²θ = 1 - (-3/5)² = 1 - 9/25 = 16/25, so sin(θ) = 4/5 (positive in Quadrant II). Choice B (-4/5) is incorrect as it has the wrong sign, forgetting that sine is positive in Quadrant II. Students must carefully apply both the Pythagorean identity and quadrant sign rules. The ASTC mnemonic reminds us that sine is positive in Quadrants I and II.
A Ferris wheel position uses $h=H+R\cos(\theta)$, where $\theta$ is an angle on the unit circle. At a checkpoint, the rotation is $\theta=120^\circ=2\pi/3$. Cosine gives the horizontal coordinate, matching the signed ratio for the height offset. Using the unit circle, what is the cosine value for $120^\circ$?
$-\sqrt{3}/2$
$-1/2$
$\sqrt{3}/2$
$1/2$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Sine and cosine values represent the coordinates on the unit circle where the angle intersects. In this problem, the angle θ is given as 120° or 2π/3 radians, requiring students to locate or calculate its trigonometric value. Choice B is correct because cos(120°) = cos(2π/3) = -1/2, which accurately reflects the cosine value for this angle in the second quadrant where cosine is negative. Choice C (-√3/2) is incorrect because it confuses the cosine value with the sine value at this angle, a common mistake when students mix up coordinates. Encourage students to memorize key unit circle values and practice identifying which coordinate corresponds to sine versus cosine. Highlight the importance of using reference angles correctly.
A satellite’s position uses $x=r\cos(\theta)$, where $\theta$ is measured from the positive $x$-axis. Suppose $\theta=330^\circ=11\pi/6$ at a particular observation time. The cosine value gives the satellite’s horizontal fraction of its orbital radius. Calculate the exact value of $\cos(\theta)$ for $\theta=11\pi/6$.
$\sqrt{3}/2$
$-\sqrt{3}/2$
$1/2$
$-1/2$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Sine and cosine values represent the coordinates on the unit circle where the angle intersects. In this problem, the angle θ is given as 330° or 11π/6 radians, requiring students to locate or calculate its trigonometric value. Choice A is correct because cos(330°) = cos(11π/6) = √3/2, which accurately reflects the cosine value for this angle in the fourth quadrant where cosine is positive. Choice B (-√3/2) is incorrect due to a quadrant sign error, a common mistake when students overlook that cosine is positive in the fourth quadrant. Encourage students to memorize key unit circle values and practice identifying quadrant signs. Highlight the importance of recognizing that cosine is positive in quadrants I and IV.
A Ferris wheel rider reaches angle $\theta=60^\circ=\pi/3$ from the center, modeling height with cosine. Using the unit circle, the radius is 1 unit and the adjacent coordinate gives $\cos\theta$. All angles are measured counterclockwise from the positive $x$-axis, with $0\le\theta\le2\pi$. Use exact trigonometric values rather than decimals. Calculate the exact value of $\cos(\theta)$ for $\theta=\pi/3$.
$-1/2$
$0.5$
$\sqrt{3}/2$
$1/2$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Cosine values represent the x-coordinate on the unit circle where the angle intersects. In this problem, the angle θ = π/3 (60°) is given, requiring students to locate its cosine value on the unit circle. Choice B (1/2) is correct because at π/3 radians, the point on the unit circle has coordinates (1/2, √3/2), making cos(π/3) = 1/2. Choice A (√3/2) is incorrect as this is the sine value at π/3, a common mistake when students confuse x and y coordinates. Students should memorize the special angles (30°, 45°, 60°) and their exact trigonometric values. Drawing the unit circle and marking these key points helps reinforce which coordinate corresponds to sine versus cosine.
A satellite is tracked from Earth’s center using a unit circle of radius 1 for direction. The line of sight makes $\theta=120^\circ=2\pi/3$ with the positive $x$-axis, placing it in Quadrant II. The $x$-coordinate equals $\cos\theta$, which must be an exact value. Degrees and radians are provided to reinforce equivalence within $0\le\theta\le2\pi$. Calculate the exact value of $\cos(\theta)$ for $\theta=2\pi/3$.
$\sqrt{3}/2$
$-\sqrt{3}/2$
$1/2$
$-1/2$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. Cosine values represent the x-coordinate on the unit circle where the angle intersects. In this problem, the angle θ = 2π/3 (120°) places the terminal side in Quadrant II, requiring students to determine both the value and sign of cosine. Choice B (-1/2) is correct because 2π/3 is the reference angle π/3 in Quadrant II, where cosine is negative, giving cos(2π/3) = -cos(π/3) = -1/2. Choice D (√3/2) is incorrect as it has the wrong sign, showing students forgot that cosine is negative in Quadrant II. Students must remember that in Quadrant II, sine is positive but cosine is negative. Using reference angles and the ASTC (All Students Take Calculus) mnemonic helps determine the correct signs.
An AC generator current is modeled as $i(t)=I\sin(\theta)$, where $\theta$ is a unit-circle angle. Measurements indicate $\cos\theta=12/13$, and the phase is in quadrant I. Using $\sin^2\theta+\cos^2\theta=1$, determine the corresponding sine ratio. Find $\sin(\theta)$ given $\cos(\theta)$ and $\theta$ is in quadrant I.
$1/13$
$-5/13$
$\sqrt{25}/13$
$5/13$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. The Pythagorean identity sin²θ + cos²θ = 1 relates sine and cosine values on the unit circle. In this problem, cos(θ) = 12/13 and θ is in quadrant I, requiring students to calculate the corresponding sine value. Choice A is correct because sin²θ = 1 - cos²θ = 1 - (12/13)² = 1 - 144/169 = 25/169, so sin(θ) = 5/13 (positive in quadrant I). Choice B (-5/13) is incorrect due to a sign error, as sine is positive in the first quadrant. Encourage students to practice using the Pythagorean identity systematically. Highlight the importance of simplifying fractions and maintaining proper signs based on quadrant location.
A pendulum’s height change can be written using $\cos(\theta)$ from a right-triangle projection. At an instant, the angle satisfies $\cos\theta=3/5$, with $\theta$ between $0^\circ$ and $90^\circ$ (quadrant I). Use the Pythagorean identity $\sin^2\theta+\cos^2\theta=1$ to relate the ratios. Based on the scenario, what is the sine of $\theta$ if $\cos\theta=3/5$?
$4/5$
$3/4$
$2/5$
$-4/5$
Explanation
This question tests AP Precalculus skills: understanding sine and cosine function values in context. The Pythagorean identity sin²θ + cos²θ = 1 relates sine and cosine values on the unit circle. In this problem, cos(θ) = 3/5 and θ is in quadrant I, requiring students to calculate the corresponding sine value. Choice A is correct because sin²θ = 1 - cos²θ = 1 - (3/5)² = 1 - 9/25 = 16/25, so sin(θ) = 4/5 (positive in quadrant I). Choice C (-4/5) is incorrect due to a sign error, as sine is positive in the first quadrant. Encourage students to practice using the Pythagorean identity and to always consider quadrant signs. Highlight the importance of checking whether the calculated value makes sense given the quadrant.