Pythagorean Identity

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Pythagorean Identity · Grade 11 · Worksheet 3

1. Aisha is analyzing the motion of a pendulum in her physics lab. She models the pendulum's position using the equation y(t) = A sin(ωt) and its velocity using v(t) = Aω cos(ωt), where A is the amplitude and ω is the angular frequency. To verify energy conservation in her model, she needs to show that for any time t, the sum of the squares of the position and velocity (scaled appropriately) remains constant. Using the trigonometric identity that relates sin²(ωt) and cos²(ωt), what should this constant sum equal? Answer: A^2 Solution: The position is y(t) = A sin(ωt) The velocity is v(t) = Aω cos(ωt) y²(t) = [A sin(ωt)]² = A² sin²(ωt) [v(t)/ω]² = [Aω cos(ωt)/ω]² = [A cos(ωt)]² = A² cos²(ωt) Adding these gives A² sin²(ωt) + A² cos²(ωt) = A²[sin²(ωt) + cos²(ωt)] Using the Pythagorean identity: sin²(ωt) + cos²(ωt) = 1 Therefore,…
   Full step-by-step solution

   Step 1: The position is y(t) = A sin(ωt) Step 2: The velocity is v(t) = Aω cos(ωt) Step 3: We need to find y²(t) + [v(t)/ω]² Step 4: y²(t) = [A sin(ωt)]² = A² sin²(ωt) Step 5: [v(t)/ω]² = [Aω cos(ωt)/ω]² = [A cos(ωt)]² = A² cos²(ωt) Step 6: Adding these gives A² sin²(ωt) + A² cos²(ωt) = A²[sin²(ωt) + cos²(ωt)] Step 7: Using the Pythagorean identity: sin²(ωt) + cos²(ωt) = 1 Step 8: Therefore, A² × 1 = A² The answer is A^2.

2. Given cos θ = 4/5, find sin²θ using the Pythagorean identity Answer: 9/25 Solution: Write the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (4/5)² = 1 Calculate (4/5)² = 16/25 Rewrite the equation: sin²θ + 16/25 = 1 Subtract 16/25 from both sides: sin²θ = 1 - 16/25 Convert 1 to 25/25: sin²θ = 25/25 - 16/25 Subtract the fractions: sin²θ = 9/25 The…
   Full step-by-step solution

   Step 1: Write the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (4/5)² = 1 Step 3: Calculate (4/5)² = 16/25 Step 4: Rewrite the equation: sin²θ + 16/25 = 1 Step 5: Subtract 16/25 from both sides: sin²θ = 1 - 16/25 Step 6: Convert 1 to 25/25: sin²θ = 25/25 - 16/25 Step 7: Subtract the fractions: sin²θ = 9/25 Step 8: The answer is sin²θ = 9/25

3. Given sin θ = 8/17, find cos θ using the identity sin²θ + cos²θ = 1 Answer: 15/17 Solution: We are given sin θ = 8/17 and need to find cos θ using the identity sin²θ + cos²θ = 1.
   Full step-by-step solution

   Step 1: We are given sin θ = 8/17 and need to find cos θ using the identity sin²θ + cos²θ = 1. Step 2: Square the given sine value: (8/17)² = 64/289 Step 3: Substitute into the identity: 64/289 + cos²θ = 1 Step 4: Solve for cos²θ: cos²θ = 1 - 64/289 = 289/289 - 64/289 = 225/289 Step 5: Take the square root: cos θ = ±√(225/289) = ±15/17 Step 6: Since no quadrant is specified, we take the positive value as the principal answer: cos θ = 15/17 The answer is 15/17.

4. Given that sin(θ) = 3/5 and θ is in the second quadrant, use the Pythagorean identity to find the exact value of cos(θ). Answer: -4/5 Solution: Recall the Pythagorean identity. sin^2(θ) + cos^2(θ) = 1 Substitute the given value of sin(θ) into the identity. We are given sin(θ) = 3/5.
   Full step-by-step solution

   Step 1: Recall the Pythagorean identity. The Pythagorean identity for sine and cosine is: sin^2(θ) + cos^2(θ) = 1 Step 2: Substitute the given value of sin(θ) into the identity. We are given sin(θ) = 3/5. Substitute this into the identity: (3/5)^2 + cos^2(θ) = 1 Step 3: Calculate (3/5)^2. (3/5)^2 = (3^2)/(5^2) = 9/25 Step 4: Rewrite the equation with the calculated value. The equation becomes: 9/25 + cos^2(θ) = 1 Step 5: Isolate cos^2(θ). Subtract 9/25 from both sides of the equation: cos^2(θ) = 1 - 9/25 Step 6: Perform the subtraction. To subtract, write 1 as a fraction with denominator 25: 1 = 25/25 So, cos^2(θ) = 25/25 - 9/25 = (25 - 9)/25 = 16/25 Step 7: Solve for cos(θ). Take the square root of both sides: cos(θ) = ±√(16/25) = ±(√16)/(√25) = ±(4/5) Step 8: Determine the correct sign using the quadrant information. We are told that θ is in the second quadrant. In the second quadrant, the x-coordinate (cosine) is negative, and the y-coordinate (sine) is positive. Since we have sin(θ) = 3/5 (positive), this confirms the second quadrant. Therefore, cosine must be negative. Step 9: State the final answer. cos(θ) = -4/5

5. Given cos θ = 9/41, find sin θ using the Pythagorean identity sin²θ + cos²θ = 1 Answer: 40/41 Solution: Write the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (9/41)² = 1 Calculate (9/41)² = 81/1681 Rewrite the equation: sin²θ + 81/1681 = 1 Subtract 81/1681 from both sides: sin²θ = 1 - 81/1681 Convert 1 to 1681/1681: sin²θ = 1681/1681 - 81/1681 Simplify: sin²θ =…
   Full step-by-step solution

   Step 1: Write the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (9/41)² = 1 Step 3: Calculate (9/41)² = 81/1681 Step 4: Rewrite the equation: sin²θ + 81/1681 = 1 Step 5: Subtract 81/1681 from both sides: sin²θ = 1 - 81/1681 Step 6: Convert 1 to 1681/1681: sin²θ = 1681/1681 - 81/1681 Step 7: Simplify: sin²θ = 1600/1681 Step 8: Take the square root: sin θ = ±√(1600/1681) = ±40/41 Step 9: Since the problem doesn't specify a quadrant, we take the positive value: sin θ = 40/41 Final answer: 40/41

6. Given sin θ = 7/25 and θ is in quadrant II, find cos θ using sin²θ + cos²θ = 1 Answer: -24/25 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (7/25)² + cos²θ = 1 Calculate (7/25)² = 49/625 Write the equation: 49/625 + cos²θ = 1 Subtract 49/625 from both sides: cos²θ = 1 - 49/625 Convert 1 to 625/625: cos²θ = 625/625 - 49/625 Simplify: cos²θ = 576/625…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (7/25)² + cos²θ = 1 Step 3: Calculate (7/25)² = 49/625 Step 4: Write the equation: 49/625 + cos²θ = 1 Step 5: Subtract 49/625 from both sides: cos²θ = 1 - 49/625 Step 6: Convert 1 to 625/625: cos²θ = 625/625 - 49/625 Step 7: Simplify: cos²θ = 576/625 Step 8: Take the square root: cos θ = ±√(576/625) = ±24/25 Step 9: Since θ is in quadrant II, cosine is negative, so cos θ = -24/25 Final answer: -24/25

7. Isabella is analyzing the orbit of a satellite using a mathematical model. She represents the satellite's position on a circular path using parametric equations: x = R cos(θ) and y = R sin(θ), where R is the orbital radius and θ is the angle from the positive x-axis. To verify that the satellite remains at a constant distance from the center of its orbit, she needs to prove that x² + y² equals a constant value for any angle θ. Using the Pythagorean identity sin²θ + cos²θ = 1, what constant value should she obtain for x² + y²? Answer: R² Solution: Write the parametric equations: x = R cos(θ), y = R sin(θ). Substitute into x² + y²: (R cos(θ))² + (R sin(θ))². Square each term: R² cos²(θ) + R² sin²(θ).
   Full step-by-step solution

   Step 1: Write the parametric equations: x = R cos(θ), y = R sin(θ). Step 2: Substitute into x² + y²: (R cos(θ))² + (R sin(θ))². Step 3: Square each term: R² cos²(θ) + R² sin²(θ). Step 4: Factor out the common factor R²: R²(cos²(θ) + sin²(θ)). Step 5: Apply the Pythagorean identity sin²θ + cos²θ = 1: R²(1). Step 6: Simplify: R². The answer is R².

8. Given cos θ = 8/17, find sin θ using sin²θ + cos²θ = 1 Answer: 15/17 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value cos θ = 8/17 into the equation: sin²θ + (8/17)² = 1 Calculate (8/17)² = 64/289 The equation becomes: sin²θ + 64/289 = 1 Subtract 64/289 from both sides: sin²θ = 1 - 64/289 Convert 1 to 289/289: sin²θ = 289/289 -…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value cos θ = 8/17 into the equation: sin²θ + (8/17)² = 1 Step 3: Calculate (8/17)² = 64/289 Step 4: The equation becomes: sin²θ + 64/289 = 1 Step 5: Subtract 64/289 from both sides: sin²θ = 1 - 64/289 Step 6: Convert 1 to 289/289: sin²θ = 289/289 - 64/289 Step 7: Simplify: sin²θ = 225/289 Step 8: Take the square root of both sides: sin θ = ±√(225/289) Step 9: Simplify: sin θ = ±15/17 Step 10: Since no quadrant is specified, we take the positive value: sin θ = 15/17 Final answer: 15/17