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Pythagorean Identity

Grade 11 · Trigonometry · Worksheet 2

  1. Mason draws a unit circle centered at the origin on a coordinate plane. He marks a point P on the circle in the third quadrant such that the y-coordinate of P is -2/sqrt(53). Using the geometric relationship between the coordinates of point P and the definitions of sine and cosine on the unit circle, prove algebraically that sin²θ + cos²θ = 1. Answer: ______________
  2. Sophia is a robotics engineer programming the arm of a manufacturing robot. The arm's endpoint moves along a circular path with radius 13 meters. At a certain moment, the vertical coordinate of the endpoint relative to the center is y = 13 sin(θ), and the horizontal coordinate is x = 13 cos(θ). The robot's control system requires that for any angle θ, the sum of the squares of the normalized coordinates equals 1. If at a particular angle, the vertical position is y = 5 meters, use the Pythagorean identity to find the exact value of the horizontal position x. Assume θ is acute.
    Answer: ______________
  3. Liam is designing a triangular support structure for a new bridge. He knows that for one particular triangle in the design, the sine of an acute angle θ is 3/5. To verify the structural integrity using the Pythagorean theorem in trigonometric form, he needs to prove the fundamental identity that relates the squares of sine and cosine for the same angle. What is the value of cos²θ for this triangle? Answer: ______________
  4. Liam is designing a triangular support structure for a bridge. He knows that for one particular triangle in his design, the ratio of the opposite side to the hypotenuse is represented by sin(θ) = 3/5. Using the Pythagorean identity, determine the exact value of cos(θ) for this triangle. Answer: ______________
  5. Given sin θ = 4/5, find cos θ using the Pythagorean identity sin²θ + cos²θ = 1 Answer: ______________
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Answer Key & Explanations

Pythagorean Identity · Grade 11 · Worksheet 2

  1. Mason draws a unit circle centered at the origin on a coordinate plane. He marks a point P on the circle in the third quadrant such that the y-coordinate of P is -2/sqrt(53). Using the geometric relationship between the coordinates of point P and the definitions of sine and cosine on the unit circle, prove algebraically that sin²θ + cos²θ = 1. Answer: 1 Solution: On a unit circle centered at the origin with radius 1, any point P = (x, y) on the circle must satisfy x² + y² = 1 by the definition of a circle.
    Full step-by-step solution

    Step 1: On a unit circle centered at the origin with radius 1, any point P = (x, y) on the circle must satisfy x² + y² = 1 by the definition of a circle. Step 2: For angle θ in standard position whose terminal side passes through P on the unit circle, cosθ = x-coordinate of P and sinθ = y-coordinate of P. Here, y = -2/sqrt(53). Step 3: Compute sin²θ = (-2/sqrt(53))² = 4/53. Step 4: Since P lies on the unit circle, x² + y² = 1. Therefore, x² = 1 - y² = 1 - 4/53 = (53/53 - 4/53) = 49/53. Step 5: Then cos²θ = x² = 49/53. Step 6: Add them: sin²θ + cos²θ = 4/53 + 49/53 = 53/53 = 1. Thus, we have proven that sin²θ + cos²θ = 1 holds for this point on the unit circle. The answer is 1.

  2. Sophia is a robotics engineer programming the arm of a manufacturing robot. The arm's endpoint moves along a circular path with radius 13 meters. At a certain moment, the vertical coordinate of the endpoint relative to the center is y = 13 sin(θ), and the horizontal coordinate is x = 13 cos(θ). The robot's control system requires that for any angle θ, the sum of the squares of the normalized coordinates equals 1. If at a particular angle, the vertical position is y = 5 meters, use the Pythagorean identity to find the exact value of the horizontal position x. Assume θ is acute. Answer: 12 Solution: We are given y = 5 and the relationship y = 13 sin(θ). So 13 sin(θ) = 5, which means sin(θ) = 5/13. Substitute sin(θ) = 5/13: (5/13)² + cos²(θ) = 1.
    Full step-by-step solution

    Step 1: We are given y = 5 and the relationship y = 13 sin(θ). So 13 sin(θ) = 5, which means sin(θ) = 5/13. Step 2: Apply the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 3: Substitute sin(θ) = 5/13: (5/13)² + cos²(θ) = 1. Step 4: Calculate (5/13)² = 25/169. So 25/169 + cos²(θ) = 1. Step 5: Subtract 25/169 from both sides: cos²(θ) = 1 - 25/169 = 169/169 - 25/169 = 144/169. Step 6: Take the positive square root (since θ is acute, cos(θ) > 0): cos(θ) = sqrt(144/169) = 12/13. Step 7: Now find x using x = 13 cos(θ): x = 13 × (12/13) = 12 meters. The answer is 12.

  3. Liam is designing a triangular support structure for a new bridge. He knows that for one particular triangle in the design, the sine of an acute angle θ is 3/5. To verify the structural integrity using the Pythagorean theorem in trigonometric form, he needs to prove the fundamental identity that relates the squares of sine and cosine for the same angle. What is the value of cos²θ for this triangle? Answer: 16/25 Solution: We are given that sin θ = 3/5 for an acute angle θ. We need to find cos² θ. sin² θ + cos² θ = 1.
    Full step-by-step solution

    We are given that sin θ = 3/5 for an acute angle θ. We need to find cos² θ. Step 1: Recall the Pythagorean identity in trigonometry: sin² θ + cos² θ = 1. Step 2: Substitute the given value of sin θ into the identity: (3/5)² + cos² θ = 1. Step 3: Calculate (3/5)²: (3/5)² = 9/25. Step 4: Substitute into the equation: 9/25 + cos² θ = 1. Step 5: Subtract 9/25 from both sides to solve for cos² θ: cos² θ = 1 - 9/25. Step 6: Write 1 as 25/25: cos² θ = 25/25 - 9/25. Step 7: Perform the subtraction: cos² θ = (25 - 9)/25 = 16/25. Thus, the value of cos² θ is 16/25.

  4. Liam is designing a triangular support structure for a bridge. He knows that for one particular triangle in his design, the ratio of the opposite side to the hypotenuse is represented by sin(θ) = 3/5. Using the Pythagorean identity, determine the exact value of cos(θ) for this triangle. Answer: 4/5 Solution: We are given that sin(θ) = 3/5. sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 3/5 into the identity: (3/5)² + cos²(θ) = 1.
    Full step-by-step solution

    We are given that sin(θ) = 3/5. Step 1: Recall the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 3/5 into the identity: (3/5)² + cos²(θ) = 1. Step 3: Calculate (3/5)²: (3/5)² = 9/25. Step 4: Substitute that into the equation: 9/25 + cos²(θ) = 1. Step 5: Subtract 9/25 from both sides to solve for cos²(θ): cos²(θ) = 1 - 9/25. Step 6: Write 1 as 25/25: cos²(θ) = 25/25 - 9/25 = 16/25. Step 7: Take the square root of both sides: cos(θ) = ±√(16/25) = ±4/5. Step 8: Determine the correct sign. Since θ is an angle in a triangular support structure for a bridge, it is an acute angle (between 0° and 90°). For acute angles, cosine is positive. Therefore, cos(θ) = 4/5. Final answer: 4/5

  5. Given sin θ = 4/5, find cos θ using the Pythagorean identity sin²θ + cos²θ = 1 Answer: 3/5 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute sin θ = 4/5: (4/5)² + cos²θ = 1 Calculate (4/5)² = 16/25: 16/25 + cos²θ = 1 Subtract 16/25 from both sides: cos²θ = 1 - 16/25 Convert 1 to 25/25: cos²θ = 25/25 - 16/25 Simplify: cos²θ = 9/25 Take the square root of both sides: cos…
    Full step-by-step solution

    Step 1: Start with 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: 16/25 + cos²θ = 1 Step 4: Subtract 16/25 from both sides: cos²θ = 1 - 16/25 Step 5: Convert 1 to 25/25: cos²θ = 25/25 - 16/25 Step 6: Simplify: cos²θ = 9/25 Step 7: Take the square root of both sides: cos θ = ±√(9/25) Step 8: Simplify: cos θ = ±3/5 Since the problem doesn't specify the quadrant, the principal value is cos θ = 3/5.