Pythagorean Identity Applications

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Answer Key & Explanations

Pythagorean Identity Applications · Grade 11 · Worksheet 1

1. Aroha is a robotics engineer programming a drone's flight stabilization algorithm. The drone's tilt angle θ relative to the horizontal is measured by an onboard gyroscope. At a particular instant, the gyroscope outputs sin(θ) = 10/13, and the control system confirms that θ is in Quadrant II. To calculate the required correction torque, Aroha needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: -sqrt(69)/13 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 10/13: (10/13)² + cos²(θ) = 1. Compute (10/13)² = 100/169.
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 10/13: (10/13)² + cos²(θ) = 1. Step 3: Compute (10/13)² = 100/169. Step 4: Rewrite the equation: 100/169 + cos²(θ) = 1. Step 5: Subtract 100/169 from both sides: cos²(θ) = 1 - 100/169. Step 6: Convert 1 to 169/169: cos²(θ) = 169/169 - 100/169. Step 7: Simplify: cos²(θ) = 69/169. Step 8: Take the square root: cos(θ) = ± sqrt(69/169) = ± sqrt(69)/13. Step 9: Since θ is in Quadrant II, cosine is negative. Therefore, cos(θ) = -sqrt(69)/13. The answer is -sqrt(69)/13.

2. If cos θ = -5/13 and θ is in quadrant II, find sin θ = ? Answer: 12/13 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (-5/13)² = 1 Calculate (-5/13)² = 25/169 The equation becomes: sin²θ + 25/169 = 1 Subtract 25/169 from both sides: sin²θ = 1 - 25/169 Calculate 1 - 25/169 = 169/169 - 25/169 = 144/169 Take the square root: sin θ…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (-5/13)² = 1 Step 3: Calculate (-5/13)² = 25/169 Step 4: The equation becomes: sin²θ + 25/169 = 1 Step 5: Subtract 25/169 from both sides: sin²θ = 1 - 25/169 Step 6: Calculate 1 - 25/169 = 169/169 - 25/169 = 144/169 Step 7: Take the square root: sin θ = ±√(144/169) = ±12/13 Step 8: Since θ is in quadrant II, sine is positive, so sin θ = 12/13 The answer is 12/13.

3. Charlotte is a civil engineer designing a curved ramp for a parking garage. The ramp's slope relative to the horizontal is modeled by an angle θ. At a specific point, an inclinometer records that sin(θ) = 7/13, and the angle θ is in Quadrant II. To calculate the horizontal force component acting on a vehicle, Charlotte needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: -sqrt(120)/13 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 7/13: (7/13)² + cos²(θ) = 1. Compute (7/13)² = 49/169.
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 7/13: (7/13)² + cos²(θ) = 1. Step 3: Compute (7/13)² = 49/169. Step 4: Rewrite the equation: 49/169 + cos²(θ) = 1. Step 5: Subtract 49/169 from both sides: cos²(θ) = 1 - 49/169. Step 6: Convert 1 to 169/169: cos²(θ) = 169/169 - 49/169. Step 7: Simplify: cos²(θ) = 120/169. Step 8: Take the square root: cos(θ) = ± sqrt(120/169) = ± sqrt(120)/13. Step 9: Since θ is in Quadrant II, cosine is negative. Therefore, cos(θ) = -sqrt(120)/13. The answer is -sqrt(120)/13.

4. If sin θ = -15/17 and θ is in quadrant III, find cos θ = ? Answer: -8/17 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (-15/17)² + cos²θ = 1 Calculate (-15/17)² = 225/289 The equation becomes: 225/289 + cos²θ = 1 Subtract 225/289 from both sides: cos²θ = 1 - 225/289 Calculate 1 - 225/289 = 289/289 - 225/289 = 64/289 Take the square root:…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (-15/17)² + cos²θ = 1 Step 3: Calculate (-15/17)² = 225/289 Step 4: The equation becomes: 225/289 + cos²θ = 1 Step 5: Subtract 225/289 from both sides: cos²θ = 1 - 225/289 Step 6: Calculate 1 - 225/289 = 289/289 - 225/289 = 64/289 Step 7: Take the square root: cos θ = ±√(64/289) = ±8/17 Step 8: Since θ is in quadrant III, cosine is negative, so cos θ = -8/17 The answer is -8/17.

5. Tane is an environmental scientist studying the flight path of a native bird. The bird's trajectory relative to a tracking station forms an angle θ with the horizontal ground. At a specific moment, the tracking system records that cos(θ) = -11/15, and the angle θ lies in Quadrant III. To calculate the bird's vertical velocity component, Tane needs the exact value of sin(θ) using the Pythagorean identity. What is the exact value of sin(θ)? Answer: -sqrt(104)/15 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute cos(θ) = -11/15: sin²(θ) + (-11/15)² = 1. Compute (-11/15)² = 121/225.
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute cos(θ) = -11/15: sin²(θ) + (-11/15)² = 1. Step 3: Compute (-11/15)² = 121/225. Step 4: Rewrite the equation: sin²(θ) + 121/225 = 1. Step 5: Subtract 121/225 from both sides: sin²(θ) = 1 - 121/225. Step 6: Convert 1 to 225/225: sin²(θ) = 225/225 - 121/225. Step 7: Simplify: sin²(θ) = 104/225. Step 8: Take the square root: sin(θ) = ± sqrt(104/225) = ± sqrt(104)/15. Step 9: Since θ is in Quadrant III, sine is negative. Therefore, sin(θ) = -sqrt(104)/15. The answer is -sqrt(104)/15.

6. Liam is designing a triangular support bracket for a robotics project. The bracket forms a right triangle where the hypotenuse is 13 cm and one acute angle θ satisfies sinθ = 5/13. He needs to calculate the exact length of the side adjacent to angle θ for precise manufacturing. What is the exact length of this adjacent side? Answer: 12 cm Solution: We have a right triangle. Hypotenuse = 13 cm. One acute angle θ satisfies sin θ = 5/13.
   Full step-by-step solution

   Let's go step-by-step. --- **Step 1: Understand the problem** We have a right triangle. Hypotenuse = 13 cm. One acute angle θ satisfies sin θ = 5/13. We need the side adjacent to angle θ. --- **Step 2: Recall definitions** In a right triangle: sin θ = opposite / hypotenuse cos θ = adjacent / hypotenuse We know sin θ = 5/13, so: opposite / hypotenuse = 5/13 opposite / 13 = 5/13 So opposite side = 5 cm. --- **Step 3: Use the Pythagorean theorem** Let adjacent side = a, opposite side = 5, hypotenuse = 13. Pythagorean theorem: a^2 + 5^2 = 13^2 a^2 + 25 = 169 a^2 = 169 - 25 a^2 = 144 a = √144 a = 12 cm. --- **Step 4: Verify using cosine** cos θ = adjacent / hypotenuse = 12/13. Check: sin^2 θ + cos^2 θ = (5/13)^2 + (12/13)^2 = 25/169 + 144/169 = 169/169 = 1. Correct. --- **Final Answer:** 12 cm

7. If sin θ = -4/5 and θ is in quadrant IV, find cos θ = ? Answer: 3/5 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute sin θ = -4/5: (-4/5)² + cos²θ = 1 Calculate (-4/5)² = 16/25 Write the equation: 16/25 + cos²θ = 1 Subtract 16/25 from both sides: cos²θ = 1 - 16/25 Calculate 1 - 16/25 = 25/25 - 16/25 = 9/25 Take square root: cos θ = ±√(9/25) = ±3/5…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute sin θ = -4/5: (-4/5)² + cos²θ = 1 Step 3: Calculate (-4/5)² = 16/25 Step 4: Write the equation: 16/25 + cos²θ = 1 Step 5: Subtract 16/25 from both sides: cos²θ = 1 - 16/25 Step 6: Calculate 1 - 16/25 = 25/25 - 16/25 = 9/25 Step 7: Take square root: cos θ = ±√(9/25) = ±3/5 Step 8: Since θ is in quadrant IV, cosine is positive, so cos θ = 3/5 The answer is 3/5.