Pythagorean Identity Applications

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Pythagorean Identity Applications · Grade 11 · Worksheet 3

1. If sin θ = -8/17 and θ is in quadrant IV, find cos θ = ? Answer: 15/17 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (-8/17)² + cos²θ = 1 Calculate (-8/17)² = 64/289 The equation becomes: 64/289 + cos²θ = 1 Subtract 64/289 from both sides: cos²θ = 1 - 64/289 Calculate 1 - 64/289 = 289/289 - 64/289 = 225/289 Take the square root: cos θ…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (-8/17)² + cos²θ = 1 Step 3: Calculate (-8/17)² = 64/289 Step 4: The equation becomes: 64/289 + cos²θ = 1 Step 5: Subtract 64/289 from both sides: cos²θ = 1 - 64/289 Step 6: Calculate 1 - 64/289 = 289/289 - 64/289 = 225/289 Step 7: Take the square root: cos θ = ±√(225/289) = ±15/17 Step 8: Since θ is in quadrant IV, cosine is positive, so cos θ = 15/17 The answer is 15/17.

2. If sin θ = -8/17 and θ is in quadrant III, find cos θ = ? Answer: -15/17 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (-8/17)² + cos²θ = 1 Calculate (-8/17)² = 64/289 The equation becomes: 64/289 + cos²θ = 1 Subtract 64/289 from both sides: cos²θ = 1 - 64/289 Calculate 1 - 64/289 = 289/289 - 64/289 = 225/289 Take the square root: cos θ…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (-8/17)² + cos²θ = 1 Step 3: Calculate (-8/17)² = 64/289 Step 4: The equation becomes: 64/289 + cos²θ = 1 Step 5: Subtract 64/289 from both sides: cos²θ = 1 - 64/289 Step 6: Calculate 1 - 64/289 = 289/289 - 64/289 = 225/289 Step 7: Take the square root: cos θ = ±√(225/289) = ±15/17 Step 8: Since θ is in quadrant III, cosine is negative, so cos θ = -15/17 The answer is -15/17.

3. Liam is designing a triangular support structure for a bridge. The structure forms a right triangle where the hypotenuse measures 13 meters and one of the acute angles is θ. Given that sin(θ) = 5/13, determine the exact value of cos(θ) and tan(θ) using the Pythagorean identity. Answer: cos(θ) = 12/13, tan(θ) = 5/12 Solution: We have a right triangle with hypotenuse 13 meters. One acute angle is θ, and sin(θ) = 5/13. Recall the definition of sine in a right triangle.
   Full step-by-step solution

   Step 1: Understand the given information. We have a right triangle with hypotenuse 13 meters. One acute angle is θ, and sin(θ) = 5/13. Step 2: Recall the definition of sine in a right triangle. sin(θ) = opposite / hypotenuse. So opposite side = 5, hypotenuse = 13. Step 3: Use the Pythagorean theorem to find the adjacent side. Let the sides be: opposite = 5, hypotenuse = 13, adjacent = a. Then: a^2 + 5^2 = 13^2 a^2 + 25 = 169 a^2 = 169 - 25 a^2 = 144 a = 12 (length is positive). Step 4: Find cos(θ). cos(θ) = adjacent / hypotenuse = 12/13. Step 5: Find tan(θ). tan(θ) = opposite / adjacent = 5/12. Step 6: Verify using the Pythagorean identity. The identity is: sin^2(θ) + cos^2(θ) = 1. sin^2(θ) = (5/13)^2 = 25/169 cos^2(θ) = (12/13)^2 = 144/169 Sum = 25/169 + 144/169 = 169/169 = 1. The identity holds. Final answer: cos(θ) = 12/13 tan(θ) = 5/12

4. A right triangle is inscribed in a unit circle such that its hypotenuse is the diameter of length 2. If one acute angle θ has a cosine value of -0.8 and the terminal side of θ lies in the second quadrant, use the Pythagorean identity to determine the exact value of sin θ. Then, find the coordinates of the point on the unit circle corresponding to angle θ. Answer: 0.6 Solution: We are given that cos θ = -0.8 and the angle is in the second quadrant.
   Full step-by-step solution

   Step 1: We are given that cos θ = -0.8 and the angle is in the second quadrant. Step 2: Apply the Pythagorean identity: sin²θ + cos²θ = 1 Step 3: Substitute the known value: sin²θ + (-0.8)² = 1 Step 4: Calculate (-0.8)² = 0.64 Step 5: The equation becomes: sin²θ + 0.64 = 1 Step 6: Subtract 0.64 from both sides: sin²θ = 1 - 0.64 = 0.36 Step 7: Take the square root: sin θ = ±√0.36 = ±0.6 Step 8: Since θ is in the second quadrant where sine is positive, sin θ = 0.6 Step 9: On the unit circle, the coordinates are (cos θ, sin θ) = (-0.8, 0.6) The exact value of sin θ is 0.6.

5. Mere is an engineer designing a robotic arm for a manufacturing plant. The arm's position is controlled by a motor that rotates through an angle θ. For a specific task, the control system requires the exact value of cos(θ) to calculate the horizontal reach of the arm. The system measures sin(θ) = 2/3, and the angle θ is in Quadrant II. Using the Pythagorean identity, what is the exact value of cos(θ)? Answer: -sqrt(5)/3 Solution: Use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1. Substitute sin(θ) = 2/3: (2/3)^2 + cos^2(θ) = 1. Compute (2/3)^2 = 4/9.
   Full step-by-step solution

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

6. Emma is analyzing the motion of a pendulum in her physics lab. The pendulum's position can be modeled by the equation y = A sin(ωt + φ), where A is the amplitude. At a specific moment, she measures that sin(ωt + φ) = 4/5. To calculate the pendulum's velocity component, she needs to find the exact value of cos(ωt + φ). What is the exact value of cos(ωt + φ)? Answer: 3/5 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (4/5)² + cos²θ = 1 Calculate (4/5)² = 16/25 So 16/25 + cos²θ = 1 Subtract 16/25 from both sides: cos²θ = 1 - 16/25 = 25/25 - 16/25 = 9/25 Take the square root: cosθ = ±√(9/25) = ±3/5 Since the pendulum's motion is…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (4/5)² + cos²θ = 1 Step 3: Calculate (4/5)² = 16/25 Step 4: So 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: cosθ = ±√(9/25) = ±3/5 Step 7: Since the pendulum's motion is typically modeled with the angle in the first quadrant where cosine is positive, we take cosθ = 3/5 The answer is 3/5.

7. Liam is designing a triangular support structure for a bridge. The angle of elevation from the base to the top is θ, and he knows that sin(θ) = 3/5. To calculate the stress distribution, he needs to find the exact value of cos(2θ). What is the value of cos(2θ)? Answer: 7/25 Solution: We are given that sin(θ) = 3/5 and need to find cos(2θ). Recall the double-angle formula for cosine. One of the formulas is: cos(2θ) = 1 - 2 sin²(θ) Substitute sin(θ) = 3/5 into the formula.
   Full step-by-step solution

   We are given that sin(θ) = 3/5 and need to find cos(2θ). Step 1: Recall the double-angle formula for cosine. One of the formulas is: cos(2θ) = 1 - 2 sin²(θ) Step 2: Substitute sin(θ) = 3/5 into the formula. cos(2θ) = 1 - 2 * (3/5)² Step 3: Calculate (3/5)². (3/5)² = 9/25 Step 4: Multiply by 2. 2 * (9/25) = 18/25 Step 5: Subtract from 1. cos(2θ) = 1 - 18/25 Step 6: Write 1 as 25/25 and subtract. 25/25 - 18/25 = 7/25 So the value of cos(2θ) is 7/25. Final answer: 7/25