Pythagorean Identity Applications Worksheets Grade 11

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)?
2. If cos θ = -5/13 and θ is in quadrant II, find sin θ = ?
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)?

…and 4 more problems

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)?
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)?
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)?

…and 3 more problems

1. If sin θ = -4/5 and θ is in quadrant III, find cos θ = ?
2. Ava is examining a point on the unit circle corresponding to an angle θ. The point lies in Quadrant II, and its x-coordinate is -1/6. Using the Pythagorean identity, find the exact value of the y-coordinate of the point.
3. Matiu is examining a point on a unit circle. The point lies in Quadrant II, and its y-coordinate is 2/3. Using the Pythagorean identity, find the exact x-coordinate of the point.

…and 5 more problems