Pythagorean Identity Applications

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Answer Key & Explanations

Pythagorean Identity Applications · Grade 11 · Worksheet 3

1. If sin θ = -4/5 and θ is in quadrant III, 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 = 25/25 - 16/25 = 9/25 Take the square root: cos θ = ±√(9/25) = ±3/5 Since θ is in…
   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 = 25/25 - 16/25 = 9/25 Step 6: Take the square root: cos θ = ±√(9/25) = ±3/5 Step 7: Since θ is in quadrant III where cosine is negative, cos θ = -3/5 The answer is -3/5.

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. Answer: √35/6 Solution: On the unit circle, coordinates are (cos θ, sin θ). Given x = -1/6, so cos θ = -1/6. Substitute cos θ = -1/6: sin²θ + (-1/6)² = 1.
   Full step-by-step solution

   Step 1: On the unit circle, coordinates are (cos θ, sin θ). Given x = -1/6, so cos θ = -1/6. Step 2: Apply the Pythagorean identity: sin²θ + cos²θ = 1. Step 3: Substitute cos θ = -1/6: sin²θ + (-1/6)² = 1. Step 4: Compute (-1/6)² = 1/36. Step 5: The equation becomes sin²θ + 1/36 = 1. Step 6: Subtract 1/36 from both sides: sin²θ = 1 - 1/36 = 36/36 - 1/36 = 35/36. Step 7: Take the square root: sin θ = ±√(35/36) = ±√35/6. Step 8: Since the point is in Quadrant II, sine is positive. Therefore, sin θ = √35/6. The exact y-coordinate is √35/6.

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. Answer: -√5/3 Solution: On the unit circle, coordinates are (cos θ, sin θ). Given y = 2/3, so sin θ = 2/3. Substitute sin θ = 2/3: (2/3)² + cos²θ = 1.
   Full step-by-step solution

   Step 1: On the unit circle, coordinates are (cos θ, sin θ). Given y = 2/3, so sin θ = 2/3. Step 2: Apply the Pythagorean identity: sin²θ + cos²θ = 1. Step 3: Substitute sin θ = 2/3: (2/3)² + cos²θ = 1. Step 4: Compute (2/3)² = 4/9. Step 5: The equation becomes 4/9 + cos²θ = 1. Step 6: Subtract 4/9 from both sides: cos²θ = 1 - 4/9 = 9/9 - 4/9 = 5/9. Step 7: Take the square root: cos θ = ±√(5/9) = ±√5/3. Step 8: Since the point is in Quadrant II, x-coordinates are negative, so cos θ = -√5/3. The exact x-coordinate is -√5/3.

4. If tan θ = 3/4 and θ is in quadrant III, find sin θ = ? Answer: -3/5 Solution: Given tan θ = 3/4, we know that sin θ / cos θ = 3/4 Let sin θ = 3k and cos θ = 4k for some constant k Apply the Pythagorean identity: sin²θ + cos²θ = 1 Substitute: (3k)² + (4k)² = 1 Calculate: 9k² + 16k² = 1 Combine: 25k² = 1 Solve for k²: k² = 1/25 Solve for k: k = ±1/5 Since θ is in quadrant…
   Full step-by-step solution

   Step 1: Given tan θ = 3/4, we know that sin θ / cos θ = 3/4 Step 2: Let sin θ = 3k and cos θ = 4k for some constant k Step 3: Apply the Pythagorean identity: sin²θ + cos²θ = 1 Step 4: Substitute: (3k)² + (4k)² = 1 Step 5: Calculate: 9k² + 16k² = 1 Step 6: Combine: 25k² = 1 Step 7: Solve for k²: k² = 1/25 Step 8: Solve for k: k = ±1/5 Step 9: Since θ is in quadrant III, both sin θ and cos θ are negative, so k must be negative: k = -1/5 Step 10: Find sin θ = 3k = 3(-1/5) = -3/5 The answer is -3/5.

5. Noah is a biomedical engineer designing a prosthetic limb that uses a hydraulic actuator to control joint movement. The actuator's force output is proportional to cos(θ), where θ is the angle of the actuator relative to the limb's frame. During a calibration test, the system measures sin(θ) = 6/11, and the angle θ is in Quadrant II. To determine the required hydraulic pressure, Noah needs the exact value of cos(θ) using the Pythagorean identity. What is the exact value of cos(θ)? Answer: -sqrt(85)/11 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 6/11: (6/11)² + cos²(θ) = 1. Compute (6/11)² = 36/121.
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 6/11: (6/11)² + cos²(θ) = 1. Step 3: Compute (6/11)² = 36/121. Step 4: Rewrite the equation: 36/121 + cos²(θ) = 1. Step 5: Subtract 36/121 from both sides: cos²(θ) = 1 - 36/121. Step 6: Convert 1 to 121/121: cos²(θ) = 121/121 - 36/121. Step 7: Simplify: cos²(θ) = 85/121. Step 8: Take the square root: cos(θ) = ± sqrt(85/121) = ± sqrt(85)/11. Step 9: Since θ is in Quadrant II, cosine is negative. Therefore, cos(θ) = -sqrt(85)/11. The answer is -sqrt(85)/11.

6. Sophia is analyzing the motion of a Ferris wheel. The height of a passenger car above the ground is modeled by a sinusoidal function, and at a certain point in time, the horizontal displacement of the car from the center of the wheel corresponds to a cosine value of -0.6. The angle θ that the car makes with the positive x-axis (center of the wheel) lies in the third quadrant. Using the Pythagorean identity, determine the exact value of sin θ. Answer: -0.8 Solution: Recall the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (-0.6)² = 1 Calculate (-0.6)² = 0.36 The equation becomes: sin²θ + 0.36 = 1 Subtract 0.36 from both sides: sin²θ = 1 - 0.36 = 0.64 Take the square root: sin θ = ±√0.64 = ±0.8 Since θ is in the third quadrant,…
   Full step-by-step solution

   Step 1: Recall the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (-0.6)² = 1 Step 3: Calculate (-0.6)² = 0.36 Step 4: The equation becomes: sin²θ + 0.36 = 1 Step 5: Subtract 0.36 from both sides: sin²θ = 1 - 0.36 = 0.64 Step 6: Take the square root: sin θ = ±√0.64 = ±0.8 Step 7: Since θ is in the third quadrant, both sine and cosine are negative, so sin θ = -0.8 The exact value of sin θ is -0.8.

7. If sin θ = -16/65 and θ is in quadrant III, find cos θ = ? Answer: -63/65 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (-16/65)² + cos²θ = 1 Calculate (-16/65)² = 256/4225 The equation becomes: 256/4225 + cos²θ = 1 Subtract 256/4225 from both sides: cos²θ = 1 - 256/4225 Calculate 1 - 256/4225 = 4225/4225 - 256/4225 = 3969/4225 Take the…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (-16/65)² + cos²θ = 1 Step 3: Calculate (-16/65)² = 256/4225 Step 4: The equation becomes: 256/4225 + cos²θ = 1 Step 5: Subtract 256/4225 from both sides: cos²θ = 1 - 256/4225 Step 6: Calculate 1 - 256/4225 = 4225/4225 - 256/4225 = 3969/4225 Step 7: Take the square root: cos θ = ±√(3969/4225) = ±63/65 Step 8: Since θ is in quadrant III, cosine is negative, so cos θ = -63/65 The answer is -63/65.

8. Emma is designing a triangular solar panel mount for a renewable energy project. The mount forms a right triangle where the hypotenuse is 17 meters and one of the acute angles is θ. If cos(θ) = 8/17, what is the exact value of sin(θ)? Answer: 15/17 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1 Substitute the given value: sin²(θ) + (8/17)² = 1 Calculate (8/17)² = 64/289 Rewrite the equation: sin²(θ) + 64/289 = 1 Subtract 64/289 from both sides: sin²(θ) = 1 - 64/289 Convert 1 to 289/289: sin²(θ) = 289/289 - 64/289 Simplify: sin²(θ) =…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1 Step 2: Substitute the given value: sin²(θ) + (8/17)² = 1 Step 3: Calculate (8/17)² = 64/289 Step 4: Rewrite the equation: sin²(θ) + 64/289 = 1 Step 5: Subtract 64/289 from both sides: sin²(θ) = 1 - 64/289 Step 6: Convert 1 to 289/289: sin²(θ) = 289/289 - 64/289 Step 7: Simplify: sin²(θ) = 225/289 Step 8: Take the square root of both sides: sin(θ) = 15/17 (since θ is acute, sin(θ) is positive) The answer is 15/17.