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Pythagorean Identity Applications

Grade 11 · Trigonometry · Worksheet 2

  1. Noah is a civil engineer analyzing the stress distribution on a curved bridge support. The load angle θ is measured from the vertical axis, and the horizontal stress component is proportional to sin(θ). At a specific load point, the sensor measures cos(θ) = -1/6, and the angle θ lies in Quadrant III. To determine the horizontal stress factor, Noah needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: ______________
  2. Mason is an engineer designing a magnetic levitation train. The vertical force component acting on a test carriage is proportional to sin(θ), where θ is the angle of the guideway relative to the horizontal. At a specific test point, the control system measures cos(θ) = -7/11, and the angle θ is in Quadrant II. To calculate the required current for the lifting electromagnets, Mason needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: ______________
  3. Matiu is a marine biologist tracking the migration of a pod of whales using sonar. The sonar system measures the angle θ of the whales' path relative to the ocean floor. At a certain moment, the system records that sin(θ) = 13/17, and the angle θ is in Quadrant I. To calculate the horizontal distance to the pod, Matiu needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: ______________
  4. Emma is a civil engineer designing a curved section of a highway. The superelevation angle θ of the road is such that the horizontal component of the normal force is proportional to cos(θ). At a particular curve, she measures sin(θ) = 9/17, and the angle θ lies in Quadrant II. To determine the friction coefficient required for safe travel, Emma needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: ______________
  5. Aroha is designing a wave energy converter that harnesses power from ocean swells. The vertical displacement of a buoy on the water's surface is given by y(t) = A sin(θ), where A is the maximum amplitude. At a specific time, the control system measures cos(θ) = -3/7, and the angle θ is in Quadrant III. To calculate the instantaneous vertical velocity of the buoy, Aroha needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: ______________
  6. If cos θ = -15/25 and θ is in quadrant II, find sin θ = ? Answer: ______________
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Answer Key & Explanations

Pythagorean Identity Applications · Grade 11 · Worksheet 2

  1. Noah is a civil engineer analyzing the stress distribution on a curved bridge support. The load angle θ is measured from the vertical axis, and the horizontal stress component is proportional to sin(θ). At a specific load point, the sensor measures cos(θ) = -1/6, and the angle θ lies in Quadrant III. To determine the horizontal stress factor, Noah needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: -sqrt(35)/6 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute cos(θ) = -1/6: sin²(θ) + (-1/6)² = 1. Compute (-1/6)² = 1/36.
    Full step-by-step solution

    Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute cos(θ) = -1/6: sin²(θ) + (-1/6)² = 1. Step 3: Compute (-1/6)² = 1/36. Step 4: Rewrite the equation: sin²(θ) + 1/36 = 1. Step 5: Subtract 1/36 from both sides: sin²(θ) = 1 - 1/36. Step 6: Convert 1 to 36/36: sin²(θ) = 36/36 - 1/36. Step 7: Simplify: sin²(θ) = 35/36. Step 8: Take the square root: sin(θ) = ± sqrt(35/36) = ± sqrt(35)/6. Step 9: Since θ is in Quadrant III, sine is negative. Therefore, sin(θ) = -sqrt(35)/6. The answer is -sqrt(35)/6.

  2. Mason is an engineer designing a magnetic levitation train. The vertical force component acting on a test carriage is proportional to sin(θ), where θ is the angle of the guideway relative to the horizontal. At a specific test point, the control system measures cos(θ) = -7/11, and the angle θ is in Quadrant II. To calculate the required current for the lifting electromagnets, Mason needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: sqrt(72)/11 = 6*sqrt(2)/11 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute cos(θ) = -7/11: sin²(θ) + (-7/11)² = 1. Compute (-7/11)² = 49/121.
    Full step-by-step solution

    Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute cos(θ) = -7/11: sin²(θ) + (-7/11)² = 1. Step 3: Compute (-7/11)² = 49/121. Step 4: Rewrite the equation: sin²(θ) + 49/121 = 1. Step 5: Subtract 49/121 from both sides: sin²(θ) = 1 - 49/121. Step 6: Convert 1 to 121/121: sin²(θ) = 121/121 - 49/121. Step 7: Simplify: sin²(θ) = 72/121. Step 8: Take the square root: sin(θ) = ± sqrt(72/121) = ± sqrt(72)/11. Step 9: Simplify sqrt(72) = sqrt(36 * 2) = 6*sqrt(2). Step 10: Since θ is in Quadrant II, sine is positive. Therefore, sin(θ) = 6*sqrt(2)/11. The answer is 6*sqrt(2)/11.

  3. Matiu is a marine biologist tracking the migration of a pod of whales using sonar. The sonar system measures the angle θ of the whales' path relative to the ocean floor. At a certain moment, the system records that sin(θ) = 13/17, and the angle θ is in Quadrant I. To calculate the horizontal distance to the pod, Matiu needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: sqrt(120)/17 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 13/17: (13/17)² + cos²(θ) = 1. Compute (13/17)² = 169/289.
    Full step-by-step solution

    Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 13/17: (13/17)² + cos²(θ) = 1. Step 3: Compute (13/17)² = 169/289. Step 4: Rewrite the equation: 169/289 + cos²(θ) = 1. Step 5: Subtract 169/289 from both sides: cos²(θ) = 1 - 169/289. Step 6: Convert 1 to 289/289: cos²(θ) = 289/289 - 169/289. Step 7: Simplify: cos²(θ) = 120/289. Step 8: Take the square root: cos(θ) = ± sqrt(120/289) = ± sqrt(120)/17. Step 9: Since θ is in Quadrant I, cosine is positive. Therefore, cos(θ) = sqrt(120)/17. The answer is sqrt(120)/17.

  4. Emma is a civil engineer designing a curved section of a highway. The superelevation angle θ of the road is such that the horizontal component of the normal force is proportional to cos(θ). At a particular curve, she measures sin(θ) = 9/17, and the angle θ lies in Quadrant II. To determine the friction coefficient required for safe travel, Emma needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: -sqrt(208)/17 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 9/17: (9/17)² + cos²(θ) = 1. Compute (9/17)² = 81/289.
    Full step-by-step solution

    Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 9/17: (9/17)² + cos²(θ) = 1. Step 3: Compute (9/17)² = 81/289. Step 4: Rewrite the equation: 81/289 + cos²(θ) = 1. Step 5: Subtract 81/289 from both sides: cos²(θ) = 1 - 81/289. Step 6: Convert 1 to 289/289: cos²(θ) = 289/289 - 81/289. Step 7: Simplify: cos²(θ) = 208/289. Step 8: Take the square root: cos(θ) = ± sqrt(208/289) = ± sqrt(208)/17. Step 9: Since θ is in Quadrant II, cosine is negative. Therefore, cos(θ) = -sqrt(208)/17. The answer is -sqrt(208)/17.

  5. Aroha is designing a wave energy converter that harnesses power from ocean swells. The vertical displacement of a buoy on the water's surface is given by y(t) = A sin(θ), where A is the maximum amplitude. At a specific time, the control system measures cos(θ) = -3/7, and the angle θ is in Quadrant III. To calculate the instantaneous vertical velocity of the buoy, Aroha needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: -sqrt(40)/7 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute cos(θ) = -3/7: sin²(θ) + (-3/7)² = 1. Compute (-3/7)² = 9/49.
    Full step-by-step solution

    Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute cos(θ) = -3/7: sin²(θ) + (-3/7)² = 1. Step 3: Compute (-3/7)² = 9/49. Step 4: Rewrite the equation: sin²(θ) + 9/49 = 1. Step 5: Subtract 9/49 from both sides: sin²(θ) = 1 - 9/49. Step 6: Convert 1 to 49/49: sin²(θ) = 49/49 - 9/49. Step 7: Simplify: sin²(θ) = 40/49. Step 8: Take the square root: sin(θ) = ± sqrt(40/49) = ± sqrt(40)/7. Step 9: Since θ is in Quadrant III, sine is negative. Therefore, sin(θ) = -sqrt(40)/7. The answer is -sqrt(40)/7.

  6. If cos θ = -15/25 and θ is in quadrant II, find sin θ = ? Answer: 20/25 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (-15/25)² = 1 Calculate (-15/25)² = 225/625 The equation becomes: sin²θ + 225/625 = 1 Subtract 225/625 from both sides: sin²θ = 1 - 225/625 Calculate 1 - 225/625 = 625/625 - 225/625 = 400/625 Take the square…
    Full step-by-step solution

    Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (-15/25)² = 1 Step 3: Calculate (-15/25)² = 225/625 Step 4: The equation becomes: sin²θ + 225/625 = 1 Step 5: Subtract 225/625 from both sides: sin²θ = 1 - 225/625 Step 6: Calculate 1 - 225/625 = 625/625 - 225/625 = 400/625 Step 7: Take the square root: sin θ = ±√(400/625) = ±20/25 Step 8: Since θ is in quadrant II, sine is positive, so sin θ = 20/25 The answer is 20/25.