Sine Cosine Graphs

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Sine Cosine Graphs · Grade 12 · Worksheet 3

1. Emma is modeling the temperature variation in a chemical reaction. The temperature, in degrees Celsius, oscillates according to a transformed sine function. The graph of the function has an amplitude of 5, a period of 3π, a phase shift of π/2 units to the right, and a vertical translation of 7 units upward. Write the equation of this sine function in the form y = A sin(B(x - C)) + D. Answer: y = 5 sin((2/3)(x - π/2)) + 7 Solution: Identify the amplitude A. The amplitude is 5, so A = 5. Find B from the period.
   Full step-by-step solution

   Step 1: Identify the amplitude A. The amplitude is 5, so A = 5. Step 2: Find B from the period. The period is 3π. For a sine function, period = 2π/B. So 2π/B = 3π. Multiply both sides by B: 2π = 3π B. Divide both sides by 3π: B = 2π / (3π) = 2/3. Step 3: Identify the phase shift C. A phase shift of π/2 units to the right means C = π/2. Step 4: Identify the vertical translation D. A translation of 7 units upward means D = 7. Step 5: Substitute all values into the general form y = A sin(B(x - C)) + D: y = 5 sin((2/3)(x - π/2)) + 7. The equation is y = 5 sin((2/3)(x - π/2)) + 7.

2. Matiu is an engineer designing a wave energy converter. The height of the water wave relative to the mean sea level is modeled by the function h(t) = 4cos(πt/6 - π/2) + 8, where h(t) is the height in meters and t is the time in seconds. Determine the exact times during the first 12 seconds when the wave height is exactly 10 meters above the mean sea level. Answer: t = 2 seconds and t = 10 seconds Solution: Set h(t) = 10. So 4cos(πt/6 - π/2) + 8 = 10. Subtract 8 from both sides: 4cos(πt/6 - π/2) = 2.
   Full step-by-step solution

   Step 1: Set h(t) = 10. So 4cos(πt/6 - π/2) + 8 = 10. Subtract 8 from both sides: 4cos(πt/6 - π/2) = 2. Divide by 4: cos(πt/6 - π/2) = 1/2. Step 2: The general solutions for cos(θ) = 1/2 are θ = π/3 + 2πk or θ = 5π/3 + 2πk, where k is an integer. So let θ = πt/6 - π/2. Step 3: First case: πt/6 - π/2 = π/3 + 2πk. Add π/2 to both sides: πt/6 = π/3 + π/2 + 2πk = (2π/6 + 3π/6) + 2πk = 5π/6 + 2πk. Multiply both sides by 6/π: t = 5 + 12k. For k = 0, t = 5 seconds. For k = 1, t = 17 seconds (outside first 12 seconds). For k = -1, t = -7 seconds (negative). So from this case, t = 5 seconds is in the interval. Step 4: Second case: πt/6 - π/2 = 5π/3 + 2πk. Add π/2: πt/6 = 5π/3 + π/2 + 2πk = (10π/6 + 3π/6) + 2πk = 13π/6 + 2πk. Multiply by 6/π: t = 13 + 12k. For k = 0, t = 13 seconds (outside first 12 seconds). For k = -1, t = 1 second. So from this case, t = 1 second is in the interval. Step 5: Check both solutions: For t = 1: h(1) = 4cos(π/6 - π/2) + 8 = 4cos(-π/3) + 8 = 4*(1/2) + 8 = 10. For t = 5: h(5) = 4cos(5π/6 - π/2) + 8 = 4cos(π/3) + 8 = 4*(1/2) + 8 = 10. Both work. The times during the first 12 seconds are t = 1 second and t = 5 seconds.

3. A transformed sine function is graphed on a coordinate plane. The wave oscillates between y = -3 and y = 5. The first peak to the right of the y-axis occurs at x = π/9, and the next peak to the right occurs at x = 7π/9. The graph passes through the midline at x = π/18, moving upward. Write the equation of this sine function in the form y = A sin(B(x - C)) + D. Answer: y = 4 sin(3(x - π/18)) + 1 Solution: Find the vertical shift D. The maximum is 5, the minimum is -3. The midline is halfway: D = (5 + (-3))/2 = 2/2 = 1.
   Full step-by-step solution

   Step 1: Find the vertical shift D. The maximum is 5, the minimum is -3. The midline is halfway: D = (5 + (-3))/2 = 2/2 = 1. Step 2: Find the amplitude A. The amplitude is the distance from midline to max: A = 5 - 1 = 4. Since the graph rises from midline upward, the sine function is positive, so A = 4. Step 3: Find the period. Two consecutive peaks are at x = π/9 and x = 7π/9. The distance between them is 7π/9 - π/9 = 6π/9 = 2π/3. This is one full period. Step 4: Find B. Period = 2π/B, so 2π/3 = 2π/B. Multiply both sides by B: (2π/3)B = 2π. Divide by 2π: B/3 = 1, so B = 3. Step 5: Find the phase shift C. The parent sine function y = sin(x) crosses the midline going upward at x = 0. In this graph, it crosses the midline going upward at x = π/18. So the graph is shifted π/18 to the right. Thus C = π/18. Step 6: Write the equation: y = 4 sin(3(x - π/18)) + 1. The answer is y = 4 sin(3(x - π/18)) + 1.

4. Liam is designing a roller coaster track that follows the path of a transformed sine function. The track's height above ground is modeled by h(t) = 3sin(π/4(t - 2)) + 5, where t is time in seconds and h(t) is height in meters. During safety testing, engineers need to determine the maximum height reached by the roller coaster and the exact times during the first complete cycle when the track reaches exactly 6.5 meters above ground. Find both the maximum height and the specific time values. Answer: Maximum height: 8 meters; Times at 6.5 meters: t = 10/3 seconds and t = 14/3 seconds Solution: Transformed sine functions follow the pattern A·sin(B(x - C)) + D, where A is amplitude, B affects the period, C is horizontal shift, and D is vertical shift. The maximum value occurs at D + A.
   Full step-by-step solution

   Transformed sine functions follow the pattern A·sin(B(x - C)) + D, where A is amplitude, B affects the period, C is horizontal shift, and D is vertical shift. The maximum value occurs at D + A. When solving trigonometric equations for specific output values, you typically isolate the trigonometric function and use inverse operations, remembering that sine functions are periodic and may have multiple solutions within one cycle.

5. y = 3sin(5(x - π/3)) + 1. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 3, Period: 2π/5, Phase shift: π/3 right, Vertical shift: 1 up Solution: Compare y = 3sin(5(x - π/3)) + 1 to the standard form y = A sin(B(x - C)) + D A = 3, so amplitude = |3| = 3 B = 5, so period = 2π/B = 2π/5 C = π/3, so phase shift = π/3 units to the right D = 1, so vertical shift = 1 unit upward Final answer: Amplitude: 3, Period: 2π/5, Phase shift: π/3 right,…
   Full step-by-step solution

   Step 1: Compare y = 3sin(5(x - π/3)) + 1 to the standard form y = A sin(B(x - C)) + D Step 2: Identify amplitude A A = 3, so amplitude = |3| = 3 Step 3: Identify period B = 5, so period = 2π/B = 2π/5 Step 4: Identify phase shift C = π/3, so phase shift = π/3 units to the right Step 5: Identify vertical shift D = 1, so vertical shift = 1 unit upward Final answer: Amplitude: 3, Period: 2π/5, Phase shift: π/3 right, Vertical shift: 1 up

6. y = 3sin(5(x - π/4)) + 1. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 3, Period: 2π/5, Phase shift: π/4 right, Vertical shift: 1 up Solution: Write the equation in standard form: y = 3sin(5(x - π/4)) + 1 Identify amplitude A = 3 Identify B = 5, so period = 2π/B = 2π/5 Identify phase shift C = π/4 (since it's (x - π/4), shift is π/4 units right) Identify vertical shift D = 1 (shift is 1 unit up) Final answer: Amplitude = 3, Period =…
   Full step-by-step solution

   Step 1: Write the equation in standard form: y = 3sin(5(x - π/4)) + 1 Step 2: Identify amplitude A = 3 Step 3: Identify B = 5, so period = 2π/B = 2π/5 Step 4: Identify phase shift C = π/4 (since it's (x - π/4), shift is π/4 units right) Step 5: Identify vertical shift D = 1 (shift is 1 unit up) Final answer: Amplitude = 3, Period = 2π/5, Phase shift = π/4 right, Vertical shift = 1 up

7. y = 5sin(3(x - π/2)) - 1. Find amplitude, period, phase shift, and vertical shift. Answer: 5, 2π/3, π/2, -1 Solution: Compare the given function y = 5sin(3(x - π/2)) - 1 to the standard form y = A sin(B(x - C)) + D Identify amplitude A = 5 Identify B = 3, so period = 2π/B = 2π/3 Identify phase shift C = π/2 (to the right) Identify vertical shift D = -1 The parameters are: amplitude = 5, period = 2π/3, phase…
   Full step-by-step solution

   Step 1: Compare the given function y = 5sin(3(x - π/2)) - 1 to the standard form y = A sin(B(x - C)) + D Step 2: Identify amplitude A = 5 Step 3: Identify B = 3, so period = 2π/B = 2π/3 Step 4: Identify phase shift C = π/2 (to the right) Step 5: Identify vertical shift D = -1 Step 6: The parameters are: amplitude = 5, period = 2π/3, phase shift = π/2, vertical shift = -1 The answer is 5, 2π/3, π/2, -1