Sine Cosine Graphs

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Sine Cosine Graphs · Grade 12 · Worksheet 3

1. A Ferris wheel at an amusement park has a diameter of 40 meters and completes one full rotation every 2 minutes. The height of a passenger above the ground can be modeled by a transformed cosine function. If the boarding platform is 5 meters above ground level and the wheel rotates counterclockwise, determine the exact height of a passenger 45 seconds after they pass the 3 o'clock position during their ride. Answer: 25 + 20√2 meters Solution: In problems involving circular motion, the height of an object can be modeled using sine or cosine functions with appropriate transformations.
   Full step-by-step solution

   In problems involving circular motion, the height of an object can be modeled using sine or cosine functions with appropriate transformations. The amplitude corresponds to the radius of the circle, the vertical shift moves the center of motion to the appropriate height, and the period relates to the time for one complete revolution. The phase shift determines where in the cycle the motion begins. When starting at a position other than the standard points, careful consideration of the function's starting value is needed.

2. y = 8sin(9(x - π/8)) + 7. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 8, Period: 2π/9, Phase shift: π/8 right, Vertical shift: 7 up Solution: Identify the parameters from the equation y = 8sin(9(x - π/8)) + 7 A = 8, B = 9, C = π/8, D = 7 Amplitude = |A| = |8| = 8 Period = 2π/|B| = 2π/9 Phase shift = C = π/8 (positive value indicates shift to the right) Vertical shift = D = 7 (positive value indicates shift upward) Final answer:…
   Full step-by-step solution

   Step 1: Identify the parameters from the equation y = 8sin(9(x - π/8)) + 7 A = 8, B = 9, C = π/8, D = 7 Step 2: Calculate amplitude Amplitude = |A| = |8| = 8 Step 3: Calculate period Period = 2π/|B| = 2π/9 Step 4: Determine phase shift Phase shift = C = π/8 (positive value indicates shift to the right) Step 5: Determine vertical shift Vertical shift = D = 7 (positive value indicates shift upward) Final answer: Amplitude: 8, Period: 2π/9, Phase shift: π/8 right, Vertical shift: 7 up

3. A cosine function is graphed on a coordinate plane. The graph shows a wave that has been vertically stretched by a factor of 2, horizontally compressed to have a period of π/2, reflected across the x-axis, shifted π/3 units to the left, and translated 1 unit downward. Write the equation of this transformed cosine function in the form y = a cos(b(x - c)) + d. Answer: y = -2 cos(4(x + π/3)) - 1 Solution: Start with the parent function y = cos(x) Apply vertical stretch by factor 2: amplitude becomes 2, so y = 2 cos(x) Apply reflection across x-axis: multiply by -1, so y = -2 cos(x) Apply horizontal compression: period becomes π/2, so b = 2π/(π/2) = 4, so y = -2 cos(4x) Apply phase shift π/3 units…
   Full step-by-step solution

   Step 1: Start with the parent function y = cos(x) Step 2: Apply vertical stretch by factor 2: amplitude becomes 2, so y = 2 cos(x) Step 3: Apply reflection across x-axis: multiply by -1, so y = -2 cos(x) Step 4: Apply horizontal compression: period becomes π/2, so b = 2π/(π/2) = 4, so y = -2 cos(4x) Step 5: Apply phase shift π/3 units left: c = -π/3, so y = -2 cos(4(x - (-π/3))) = -2 cos(4(x + π/3)) Step 6: Apply vertical translation 1 unit down: d = -1, so y = -2 cos(4(x + π/3)) - 1 Step 7: Final equation: y = -2 cos(4(x + π/3)) - 1

4. A sine function is graphed on a coordinate plane with amplitude 3, period π, phase shift π/4 to the right, and vertical translation 2 units upward. The function passes through the point (π/2, 5). Write the equation of this sine function in the form y = a sin(b(x - c)) + d. Answer: y = 3 sin(2(x - π/4)) + 2 Solution: Amplitude = 3 Period = π Phase shift = π/4 to the right Vertical translation = 2 units upward Point on graph: (π/2, 5) y = a sin(b(x - c)) + d Amplitude = |a| = 3 So a = 3 (positive unless other info says otherwise).
   Full step-by-step solution

   Let's go step by step. We are given: Amplitude = 3 Period = π Phase shift = π/4 to the right Vertical translation = 2 units upward Point on graph: (π/2, 5) The general form is: y = a sin(b(x - c)) + d --- **Step 1: Determine a (amplitude)** Amplitude = |a| = 3 So a = 3 (positive unless other info says otherwise). --- **Step 2: Determine b (from period)** Period = 2π / |b| = π So 2π / |b| = π Multiply both sides by |b|: 2π = π |b| Divide by π: 2 = |b| So b = 2 or b = -2. Usually we take b positive unless phase shift direction changes. Let's take b = 2 for now. --- **Step 3: Determine c (phase shift)** Phase shift π/4 to the right means c = π/4. Because in y = a sin(b(x - c)) + d, phase shift = c to the right. --- **Step 4: Determine d (vertical shift)** Vertical translation 2 units upward means d = 2. --- So far we have: y = 3 sin(2(x - π/4)) + 2 --- **Step 5: Check if it passes through (π/2, 5)** Plug x = π/2: y = 3 sin(2(π/2 - π/4)) + 2 = 3 sin(2(π/4)) + 2 = 3 sin(π/2) + 2 = 3(1) + 2 = 5 Yes, it works. So b = 2 is correct. --- **Step 6: Final equation** y = 3 sin(2(x - π/4)) + 2 --- ANSWER: y = 3 sin(2(x - π/4)) + 2

5. y = 7sin(2(x - π/2)) - 3. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 7, Period: π, Phase shift: π/2 right, Vertical shift: 3 down Solution: Write the equation in standard form: y = 7sin(2(x - π/2)) - 3 Identify A (amplitude): A = 7 Identify B (affects period): B = 2 Calculate period: Period = 2π/B = 2π/2 = π Identify C (phase shift): C = π/2 Determine phase shift direction: Since it's (x - π/2), the shift is π/2 units to the right…
   Full step-by-step solution

   Step 1: Write the equation in standard form: y = 7sin(2(x - π/2)) - 3 Step 2: Identify A (amplitude): A = 7 Step 3: Identify B (affects period): B = 2 Step 4: Calculate period: Period = 2π/B = 2π/2 = π Step 5: Identify C (phase shift): C = π/2 Step 6: Determine phase shift direction: Since it's (x - π/2), the shift is π/2 units to the right Step 7: Identify D (vertical shift): D = -3, so vertical shift is 3 units down Step 8: Final answer: Amplitude = 7, Period = π, Phase shift = π/2 right, Vertical shift = 3 down

6. A marine biologist is studying the vertical motion of a dolphin swimming near the surface. The dolphin's depth below the water surface follows the function d(t) = 3sin(πt/4 - π/2) + 2, where d is depth in meters and t is time in seconds. Determine the maximum depth the dolphin reaches below the surface and the time when it first reaches this maximum depth after t = 0. Answer: 5 meters at t = 6 seconds Solution: When analyzing transformed trigonometric functions of the form A·sin(B(x - C)) + D, the amplitude |A| determines the vertical stretch, while the vertical shift D moves the midline. The maximum value occurs when the sine function reaches its peak of 1.
   Full step-by-step solution

   When analyzing transformed trigonometric functions of the form A·sin(B(x - C)) + D, the amplitude |A| determines the vertical stretch, while the vertical shift D moves the midline. The maximum value occurs when the sine function reaches its peak of 1. Phase shifts affect the horizontal positioning of these extreme points. In real-world applications like wave motion or periodic behavior, these transformations correspond to physical characteristics such as maximum displacement and timing of events.

7. A marine biologist is studying the vertical motion of a dolphin swimming near a buoy. The dolphin's depth below the surface is modeled by the function d(t) = 4sin(πt/3 - π/2) + 6, where d is depth in meters and t is time in seconds. Determine the maximum depth the dolphin reaches and the first time after t=0 when it reaches this maximum depth. Answer: 10 meters at t=3 seconds Solution: d(t) = 4 sin(πt/3 - π/2) + 6 The sine function sin(x) has a range [-1, 1]. So 4 sin(πt/3 - π/2) has a range [-4, 4]. Depth d(t) = 4 sin(πt/3 - π/2) + 6.
   Full step-by-step solution

   Let's work through this step by step. We are given: d(t) = 4 sin(πt/3 - π/2) + 6 --- **Step 1: Understanding the sine function range** The sine function sin(x) has a range [-1, 1]. So 4 sin(πt/3 - π/2) has a range [-4, 4]. --- **Step 2: Finding the maximum depth** Depth d(t) = 4 sin(πt/3 - π/2) + 6. The maximum of d(t) occurs when sin(πt/3 - π/2) is maximum. Maximum of sine = 1. So maximum d(t) = 4 * 1 + 6 = 10 meters. --- **Step 3: When does the maximum occur?** We need sin(πt/3 - π/2) = 1. We know sin(θ) = 1 when θ = π/2 + 2πk, where k is an integer. So: πt/3 - π/2 = π/2 + 2πk --- **Step 4: Solve for t** πt/3 - π/2 = π/2 + 2πk Add π/2 to both sides: πt/3 = π + 2πk Divide both sides by π: t/3 = 1 + 2k Multiply by 3: t = 3 + 6k, where k is an integer. --- **Step 5: Find the first time after t=0** k = 0 gives t = 3 seconds. k = -1 gives t = -3 seconds (before t=0). So the first time after t=0 when the dolphin reaches maximum depth is t = 3 seconds. --- **Final answer:** Maximum depth = 10 meters at t = 3 seconds.