This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically phase shift from horizontal translations. For cos(t - k), the phase shift is k units to the right, as the function reaches any given value k units later than the standard cosine. In V(t) = cos(t - π/4), the expression (t - π/4) indicates a horizontal shift of π/4 units to the right. Choice B correctly identifies this as a right shift of π/4, showing the cosine wave starts π/4 units later than usual. Choice A would be correct if the sign were positive inside the parentheses, while choice D fails to recognize the transformation. To help students: remember that f(t - h) shifts right by h units, consistent with general function transformations. Verify phase shifts by checking specific points, such as where the function equals its maximum value.

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically calculating period with fractional coefficients. The period of cos(Bt) is 2π/B, and when B is a fraction, the period becomes larger than the standard 2π. For V(t) = 2cos(t/3), we can rewrite as 2cos((1/3)t), so B = 1/3, giving period = 2π/(1/3) = 6π. Choice B correctly identifies the period as 6π, showing that the function takes three times longer to complete one cycle compared to standard cosine. Choice C incorrectly inverts the relationship, while choice A misapplies the coefficient 3. To help students: practice rewriting expressions like cos(t/n) as cos((1/n)t) to clearly identify B. Emphasize that dividing t by a number stretches the graph horizontally, increasing the period.

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically how the coefficient of t affects the period. The standard cosine function cos(t) has period 2π, but cos(Bt) has period 2π/B, so larger B values create shorter periods and more rapid oscillations. For V(t) = cos(2t), B = 2, giving period = 2π/2 = π, which is half the standard period of 2π. Choice B correctly states that the period halves, as the function now completes two full cycles in the time it previously took for one cycle. Choice A incorrectly suggests the period doubles, reversing the relationship between B and period, while choice D misses the transformation entirely. To help students: demonstrate graphically how cos(2t) compresses the standard cosine horizontally. Practice with various B values to build intuition that B > 1 compresses (shorter period) while 0 < B < 1 stretches (longer period).

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically period changes. The period of y = cos(Bx) is calculated using period = 2π/|B|, where B is the coefficient of x inside the cosine function. For y = cos(3x), B = 3, so period = 2π/3, meaning the circuit completes three full cycles in the space of 2π radians. Choice C correctly identifies the period as 2π/3, showing the function oscillates three times faster than standard cosine. Choice A incorrectly suggests 6π; Choice B suggests 3π; Choice D suggests π/3, all misapplying the period formula. To help students: consistently use period = 2π/|B| and remember that the coefficient B tells you how many complete cycles occur in 2π radians. Verify by checking that cos(3(x + 2π/3)) = cos(3x + 2π) = cos(3x).

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically determining period from the equation. The period of y = sin(Bx) is calculated using the formula period = 2π/|B|, where B is the coefficient of x inside the sine function. For y = sin(4x), B = 4, so the period = 2π/4 = π/2, meaning the function completes one full cycle in π/2 radians. Choice B correctly identifies the period as π/2, showing the function oscillates four times faster than standard sine. Choice A incorrectly suggests 8π; Choice C suggests 4π; Choice D suggests the standard 2π period, all failing to apply the period formula correctly. To help students: consistently apply the formula period = 2π/|B| and remember that larger B values create smaller periods (faster oscillations). Verify by checking that sin(4(x + π/2)) = sin(4x + 2π) = sin(4x).

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically period changes with fractional coefficients. The period formula is period = 2π/|B| for y = sin(Bx), where smaller B values create larger periods (slower oscillations). For y = sin(1/2·x), B = 1/2, so period = 2π/(1/2) = 2π × 2 = 4π, meaning the pendulum takes twice as long to complete one cycle. Choice B correctly identifies the period as 4π, showing the function oscillates half as fast as standard sine. Choice A incorrectly suggests π; Choice C suggests the standard 2π; Choice D confuses period with the coefficient. To help students: when B is a fraction, remember that dividing by a fraction means multiplying by its reciprocal. Visualize that smaller B values stretch the graph horizontally.

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically period changes. The period of a trigonometric function is determined by the coefficient of x inside the function, where period = 2π/|B| for y = cos(Bx). For y = cos(2x), B = 2, so the period = 2π/2 = π, which is half the standard period of 2π. Choice B correctly identifies the period as π, showing that when the coefficient inside is 2, the function completes its cycle twice as fast. Choice A incorrectly suggests the period doubles; Choice C incorrectly states it stays the same; Choice D confuses period with the coefficient. To help students: use the formula period = 2π/|B| consistently and visualize how larger B values compress the graph horizontally. Practice with various coefficients and verify by graphing to see complete cycles.

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically amplitude. The amplitude of a trigonometric function is the absolute value of the coefficient in front of the sine or cosine function, determining the maximum distance from the midline to the peak or trough. For the equation y = 3sin(x), the coefficient is 3, so the amplitude is |3| = 3, meaning the wave oscillates between -3 and 3. Choice B correctly identifies the amplitude as 3, which matches the coefficient of the sine function. Choice A incorrectly suggests 1/3, possibly confusing reciprocal relationships; Choice C incorrectly uses 2π, which is the period of standard sine; Choice D incorrectly uses -3, failing to take the absolute value. To help students: emphasize that amplitude is always positive and equals the absolute value of the coefficient. Practice identifying transformations by comparing equations to the standard form y = A·sin(Bx + C) + D, where |A| is the amplitude.

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically identifying amplitude. In the general form A·cos(Bt + C) + D, the amplitude is |A|, which represents the maximum distance from the midline to the peak or trough of the function. For the equation I(t) = 10cos(60πt), the coefficient of cosine is 10, making the amplitude |10| = 10. Choice B correctly identifies the amplitude as 10, which determines how far the current oscillates above and below its center value. Choice A incorrectly uses 60π, confusing the frequency coefficient with amplitude, while choices C and D misinterpret other components of the equation. To help students: emphasize that amplitude is always the absolute value of the coefficient directly multiplying the trigonometric function. Encourage students to identify each transformation parameter systematically by comparing to the standard form.

This question tests AP Precalculus skills: understanding sine and cosine function transformations, specifically calculating the period. The period of cos(Bt) is 2π/B, where B is the coefficient of t inside the cosine function, determining how quickly the function completes one full cycle. For I(t) = 5cos(4πt), B = 4π, so the period is 2π/(4π) = 1/2, which equals π/2. Choice B correctly identifies the period as π/2, representing the time for one complete oscillation of the alternating current. Choice A incorrectly calculates 4π, possibly confusing period with the coefficient, while choice C gives the standard period without accounting for the frequency change. To help students: practice the period formula systematically and verify by checking that cos(B(t + period)) = cos(Bt). Watch for errors in algebraic simplification when dividing by coefficients containing π.