Sine Cosine Graphs

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Sine Cosine Graphs · Grade 12 · Worksheet 1

1. An oceanographer is studying the vertical motion of a research buoy in wavy conditions. The buoy's height above its equilibrium position follows the function h(t) = 2sin(πt/3) + 3cos(πt/3), where h is height in meters and t is time in seconds. Determine the exact maximum height the buoy reaches above its equilibrium position during its motion. Answer: √13 Solution: For functions of the form A sin(ωt) + B cos(ωt), the maximum value can be found by treating it as a single sine or cosine function with a phase shift.
   Full step-by-step solution

   For functions of the form A sin(ωt) + B cos(ωt), the maximum value can be found by treating it as a single sine or cosine function with a phase shift. The amplitude of the resulting function is determined by the square root of the sum of the squares of the coefficients. This concept applies to many oscillatory systems in physics and engineering.

2. ∫(3x² - 2x + 1) dx from 0 to 2 = ? Answer: 6 Solution: ∫ from 0 to 2 of (3x² - 2x + 1) dx ∫ 3x² dx = 3 * (x³/3) = x³ ∫ -2x dx = -2 * (x²/2) = -x² ∫ 1 dx = x F(x) = x³ - x² + x The definite integral = F(2) - F(0) First, compute F(2): F(2) = (2)³ - (2)² + (2) = 8 - 4 + 2 = 6 Next, compute F(0): F(0) = (0)³ - (0)² + (0) = 0 F(2) - F(0) = 6 - 0 = 6…
   Full step-by-step solution

   Let's solve the definite integral step by step. We are given: ∫ from 0 to 2 of (3x² - 2x + 1) dx --- **Step 1: Find the antiderivative** For each term: ∫ 3x² dx = 3 * (x³/3) = x³ ∫ -2x dx = -2 * (x²/2) = -x² ∫ 1 dx = x So the antiderivative F(x) is: F(x) = x³ - x² + x --- **Step 2: Apply the Fundamental Theorem of Calculus** The definite integral = F(2) - F(0) First, compute F(2): F(2) = (2)³ - (2)² + (2) = 8 - 4 + 2 = 6 Next, compute F(0): F(0) = (0)³ - (0)² + (0) = 0 --- **Step 3: Subtract** F(2) - F(0) = 6 - 0 = 6 --- **Final Answer:** 6

3. A cosine function is graphed on a coordinate plane. The function has been transformed from the parent function y = cos(x) by a vertical stretch of factor 4, a horizontal compression such that the period becomes 2π/3, a phase shift of π/6 units to the left, and a vertical translation of 1 unit downward. Write the equation of the transformed function in the form y = a cos(b(x - c)) + d. Answer: y = 4 cos(3(x + π/6)) - 1 Solution: Identify the amplitude (a). A vertical stretch of factor 4 means a = 4. Determine the horizontal compression factor (b).
   Full step-by-step solution

   Step 1: Identify the amplitude (a). A vertical stretch of factor 4 means a = 4. Step 2: Determine the horizontal compression factor (b). The period of the parent function is 2π. The new period is 2π/3. Using the formula period = 2π/b, we have 2π/b = 2π/3. Solving for b: b = 3. Step 3: Determine the phase shift (c). A phase shift of π/6 units to the left means c = -π/6. In the form y = a cos(b(x - c)), this becomes (x - (-π/6)) = (x + π/6). Step 4: Determine the vertical translation (d). A vertical translation of 1 unit downward means d = -1. Step 5: Combine all parameters: a = 4, b = 3, c = -π/6, d = -1. Step 6: Write the final equation: y = 4 cos(3(x + π/6)) - 1.

4. Aroha is an astronomer studying the light intensity of a variable star. The star's brightness over time is modeled by the function I(t) = 12 sin(π/9 (t - 3)) + 15, where I(t) is the brightness in arbitrary units and t is the time in days. Determine the maximum brightness of the star and the first time after t = 0 when this maximum occurs. Answer: Maximum brightness: 27 units; first maximum at t = 7.5 days Solution: Identify parameters: A = 12, B = π/9, C = 3, D = 15. The maximum of sin(B(x-C)) is 1, so maximum I(t) = 12(1) + 15 = 27 units.
   Full step-by-step solution

   Step 1: Identify parameters: A = 12, B = π/9, C = 3, D = 15. Step 2: The maximum of sin(B(x-C)) is 1, so maximum I(t) = 12(1) + 15 = 27 units. Step 3: To find when this first occurs after t=0, set the argument equal to π/2: (π/9)(t - 3) = π/2 Multiply both sides by 9/π: t - 3 = (9/π)(π/2) = 9/2 = 4.5 So t = 4.5 + 3 = 7.5 days. Step 4: Verify this is the first positive time: The period is 2π/(π/9) = 18 days. The sine function reaches its maximum at π/2, 5π/2, etc. The next solution would be (π/9)(t - 3) = 5π/2 → t - 3 = 22.5 → t = 25.5 days, which is later. So t = 7.5 days is the first maximum after t=0. Final answer: Maximum brightness = 27 units; first maximum at t = 7.5 days.

5. Noah is a marine engineer designing an underwater monitoring station. The station's vertical position relative to the ocean floor is controlled by a buoyancy system that follows a transformed sine function. The height h(t) in meters above the ocean floor is given by h(t) = 6 sin(π/6 (t - 1)) + 11, where t is time in seconds. Determine the maximum height of the station above the ocean floor and the first time after t = 0 seconds when this maximum height occurs. Answer: Maximum height = 17 meters, first time = 4 seconds Solution: Identify parameters. The function is h(t) = 6 sin(π/6 (t - 1)) + 11. Amplitude A = 6, period = 2π / B = 2π / (π/6) = 12 seconds, horizontal shift C = 1 second to the right, vertical shift D = 11 meters.
   Full step-by-step solution

   Step 1: Identify parameters. The function is h(t) = 6 sin(π/6 (t - 1)) + 11. Amplitude A = 6, period = 2π / B = 2π / (π/6) = 12 seconds, horizontal shift C = 1 second to the right, vertical shift D = 11 meters. Step 2: The sine function ranges from -1 to 1, so the minimum of sin(...) is -1 and maximum is 1. Step 3: Maximum height occurs when sin(π/6 (t - 1)) = 1. Then h_max = 6(1) + 11 = 17 meters. Step 4: Solve sin(π/6 (t - 1)) = 1. The sine function equals 1 at π/2 + 2πk for integer k. So π/6 (t - 1) = π/2 + 2πk. Multiply both sides by 6/π: t - 1 = 3 + 12k. So t = 4 + 12k. Step 5: For the first time after t = 0, set k = 0, giving t = 4 seconds. Final answer: Maximum height = 17 meters, first time = 4 seconds.

6. y = 3sin(2(x - π/4)) + 1. Identify amplitude, period, phase shift, and vertical shift. Answer: Amplitude: 3, Period: π, Phase shift: π/4 right, Vertical shift: 1 up Solution: Compare y = 3sin(2(x - π/4)) + 1 to standard form y = A sin(B(x - C)) + D Identify amplitude A = 3 Identify B = 2, so period = 2π/B = 2π/2 = π 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:…
   Full step-by-step solution

   Step 1: Compare y = 3sin(2(x - π/4)) + 1 to standard form y = A sin(B(x - C)) + D Step 2: Identify amplitude A = 3 Step 3: Identify B = 2, so period = 2π/B = 2π/2 = π 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: π, Phase shift: π/4 right, Vertical shift: 1 up

7. A sine wave 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/4, 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: Sine function transformations follow specific patterns: amplitude multiplies the output, period relates to the horizontal stretch, phase shift moves the graph left or right, and vertical translation shifts the entire graph up or down. The general form y = a sin(b(x - c)) + d helps organize these…
   Full step-by-step solution

   Sine function transformations follow specific patterns: amplitude multiplies the output, period relates to the horizontal stretch, phase shift moves the graph left or right, and vertical translation shifts the entire graph up or down. The general form y = a sin(b(x - c)) + d helps organize these transformations systematically.