Pythagorean Identity Applications

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Answer Key & Explanations

Pythagorean Identity Applications · Grade 11 · Worksheet 1

1. 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.

2. A right triangle is inscribed in a unit circle such that its hypotenuse is the diameter. The point on the unit circle corresponding to angle θ has coordinates (x, y). If sin(θ) = 3/5 and the terminal side of θ lies in the second quadrant, use the Pythagorean identity to determine the exact value of cos(θ). Answer: -4/5 Solution: Recall the Pythagorean identity: sin²(θ) + cos²(θ) = 1 Substitute the given value: (3/5)² + cos²(θ) = 1 Calculate (3/5)² = 9/25 Write the equation: 9/25 + cos²(θ) = 1 Subtract 9/25 from both sides: cos²(θ) = 1 - 9/25 Calculate 1 - 9/25 = 25/25 - 9/25 = 16/25 Take the square root: cos(θ) =…
   Full step-by-step solution

   Step 1: Recall the Pythagorean identity: sin²(θ) + cos²(θ) = 1 Step 2: Substitute the given value: (3/5)² + cos²(θ) = 1 Step 3: Calculate (3/5)² = 9/25 Step 4: Write the equation: 9/25 + cos²(θ) = 1 Step 5: Subtract 9/25 from both sides: cos²(θ) = 1 - 9/25 Step 6: Calculate 1 - 9/25 = 25/25 - 9/25 = 16/25 Step 7: Take the square root: cos(θ) = ±√(16/25) = ±4/5 Step 8: Since θ is in the second quadrant, cosine is negative, so cos(θ) = -4/5 The answer is -4/5.

3. A right triangle is inscribed in a unit circle such that its hypotenuse is the diameter of length 2. If one of the acute angles is θ, and the side opposite to θ has length √3, use the Pythagorean identity to determine the exact value of cos(θ). Answer: 1/2 Solution: 1. The hypotenuse length is 2 (since unit circle radius = 1, diameter = 2). One acute angle is θ, and the side opposite θ has length √3.
   Full step-by-step solution

   Step-by-step solution: 1. Understand the problem: A right triangle is inscribed in a unit circle with hypotenuse as the diameter. The hypotenuse length is 2 (since unit circle radius = 1, diameter = 2). One acute angle is θ, and the side opposite θ has length √3. 2. Label the triangle: Hypotenuse = 2 Side opposite θ = √3 Let the side adjacent to θ = x 3. Apply the Pythagorean theorem: (opposite)^2 + (adjacent)^2 = (hypotenuse)^2 (√3)^2 + x^2 = 2^2 4. Calculate: 3 + x^2 = 4 x^2 = 4 - 3 x^2 = 1 x = 1 (length is positive) 5. Now we know: Adjacent side to θ = 1 Hypotenuse = 2 6. Use definition of cosine: cos(θ) = adjacent/hypotenuse cos(θ) = 1/2 7. Verify using Pythagorean identity: sin(θ) = opposite/hypotenuse = √3/2 Pythagorean identity: sin^2(θ) + cos^2(θ) = 1 (√3/2)^2 + cos^2(θ) = 1 3/4 + cos^2(θ) = 1 cos^2(θ) = 1 - 3/4 cos^2(θ) = 1/4 cos(θ) = 1/2 (positive since θ is acute) Final answer: cos(θ) = 1/2

4. 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.

5. Matiu is analyzing the oscillations of a suspension bridge in a wind tunnel. The displacement of a sensor on the bridge deck is modeled by d(t) = A cos(θ), where A is the maximum amplitude. At a specific moment, the sensor records sin(θ) = 9/41. To calculate the horizontal acceleration of the deck, Matiu needs the exact value of cos(θ) using the Pythagorean identity, given that θ is in Quadrant IV. What is the exact value of cos(θ)? Answer: sqrt(1600)/41 = 40/41 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 9/41: (9/41)² + cos²(θ) = 1. Compute (9/41)² = 81/1681.
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 9/41: (9/41)² + cos²(θ) = 1. Step 3: Compute (9/41)² = 81/1681. Step 4: Rewrite: 81/1681 + cos²(θ) = 1. Step 5: Subtract 81/1681 from both sides: cos²(θ) = 1 - 81/1681. Step 6: Convert 1 to 1681/1681: cos²(θ) = 1681/1681 - 81/1681. Step 7: Simplify: cos²(θ) = 1600/1681. Step 8: Take the square root: cos(θ) = ± sqrt(1600/1681) = ± 40/41. Step 9: Since θ is in Quadrant IV, cosine is positive. Therefore, cos(θ) = 40/41. The answer is 40/41.

6. 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.

7. Sophia is designing a sound wave model for a physics experiment. The displacement of a particle in the medium is given by y = A sin(θ), where A is the amplitude. At a particular instant, she determines that cos(θ) = 1/6. To calculate the particle's velocity, she needs the exact value of sin(θ) using the Pythagorean identity, given that θ is in Quadrant II. What is the exact value of sin(θ)? Answer: -sqrt(35)/6 Solution: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Substitute cos(θ) = 1/6: sin²(θ) + (1/6)² = 1. Compute (1/6)² = 1/36.
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute cos(θ) = 1/6: sin²(θ) + (1/6)² = 1. Step 3: Compute (1/6)² = 1/36. Step 4: Rewrite: sin²(θ) + 1/36 = 1. Step 5: Subtract 1/36 from both sides: sin²(θ) = 1 - 1/36. Step 6: Convert 1 to 36/36: sin²(θ) = 36/36 - 1/36. Step 7: Simplify: sin²(θ) = 35/36. Step 8: Take the square root: sin(θ) = ± sqrt(35)/6. Step 9: Since θ is in Quadrant II, sin(θ) is positive. Therefore, sin(θ) = sqrt(35)/6. The answer is sqrt(35)/6.