Pythagorean Identity Applications

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Pythagorean Identity Applications · Grade 11 · Worksheet 2

1. Liam is designing a triangular support bracket for a robotics project. The bracket forms a right triangle where the hypotenuse is 13 cm and one of the acute angles is θ. If sin(θ) = 5/13, what is the exact value of cos(θ)? Answer: 12/13 Solution: We are given a right triangle with hypotenuse 13 cm and an acute angle θ such that sin(θ) = 5/13. Recall the definition of sine in a right triangle. sin(θ) = opposite / hypotenuse.
   Full step-by-step solution

   We are given a right triangle with hypotenuse 13 cm and an acute angle θ such that sin(θ) = 5/13. Step 1: Recall the definition of sine in a right triangle. sin(θ) = opposite / hypotenuse. Here, sin(θ) = 5/13 means the side opposite θ is 5 and the hypotenuse is 13. Step 2: Find the length of the adjacent side using the Pythagorean theorem. Let the sides be: opposite = 5, hypotenuse = 13, adjacent = a. We have: a^2 + 5^2 = 13^2 a^2 + 25 = 169 a^2 = 169 - 25 a^2 = 144 a = 12 (length is positive) Step 3: Use the definition of cosine in a right triangle. cos(θ) = adjacent / hypotenuse cos(θ) = 12 / 13 Step 4: Conclusion. The exact value of cos(θ) is 12/13.

2. If sin θ = -7/25 and θ is in quadrant IV, find cos θ = ? Answer: 24/25 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (-7/25)² + cos²θ = 1 Calculate (-7/25)² = 49/625 The equation becomes: 49/625 + cos²θ = 1 Subtract 49/625 from both sides: cos²θ = 1 - 49/625 Calculate 1 - 49/625 = 625/625 - 49/625 = 576/625 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: (-7/25)² + cos²θ = 1 Step 3: Calculate (-7/25)² = 49/625 Step 4: The equation becomes: 49/625 + cos²θ = 1 Step 5: Subtract 49/625 from both sides: cos²θ = 1 - 49/625 Step 6: Calculate 1 - 49/625 = 625/625 - 49/625 = 576/625 Step 7: Take the square root: cos θ = ±√(576/625) = ±24/25 Step 8: Since θ is in quadrant IV, cosine is positive, so cos θ = 24/25 The answer is 24/25.

3. If sin θ = -12/13 and θ is in quadrant IV, find cos θ = ? Answer: 5/13 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (-12/13)² + cos²θ = 1 Calculate (-12/13)² = 144/169 Write the equation: 144/169 + cos²θ = 1 Subtract 144/169 from both sides: cos²θ = 1 - 144/169 Calculate 1 - 144/169 = 169/169 - 144/169 = 25/169 Take the square root:…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (-12/13)² + cos²θ = 1 Step 3: Calculate (-12/13)² = 144/169 Step 4: Write the equation: 144/169 + cos²θ = 1 Step 5: Subtract 144/169 from both sides: cos²θ = 1 - 144/169 Step 6: Calculate 1 - 144/169 = 169/169 - 144/169 = 25/169 Step 7: Take the square root: cos θ = ±√(25/169) = ±5/13 Step 8: Since θ is in quadrant IV, cosine is positive, so cos θ = 5/13 The answer is 5/13.

4. Noah is modeling the trajectory of a particle in a magnetic field. The particle's position vector makes an angle θ with the vertical axis, and he determines that sin(θ) = 1/6. To calculate the particle's horizontal velocity component, he needs the exact value of cos(θ) using the Pythagorean identity, given that θ is in Quadrant I. What is the exact value of cos(θ)? Answer: sqrt(35)/6 Solution: Use the Pythagorean identity: sin^2(θ) + cos^2(θ) = 1. Substitute sin(θ) = 1/6: (1/6)^2 + cos^2(θ) = 1. Compute (1/6)^2 = 1/36.
   Full step-by-step solution

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

5. Liam is designing a suspension bridge where the main cable forms a parabolic shape. At a point 50 meters horizontally from the lowest point of the cable, the cable makes an angle of 30° with the horizontal. If the cable's slope at any point is given by tan(θ), use the Pythagorean identity to find the exact value of sin(θ) at this location. Answer: 1/2 Solution: We are told that at a point 50 meters horizontally from the lowest point of the cable, the cable makes an angle of 30° with the horizontal.
   Full step-by-step solution

   We are told that at a point 50 meters horizontally from the lowest point of the cable, the cable makes an angle of 30° with the horizontal. The slope of the cable is given by tan(θ), and we are to find sin(θ) at this location. Step 1: Interpret the given angle. The problem says: "the cable makes an angle of 30° with the horizontal." That means θ = 30° at this location. Step 2: Use the Pythagorean identity to find sin(θ) from tan(θ). We know tan(30°) = 1/√3. The identity relating sine and tangent is: tan(θ) = sin(θ) / cos(θ) Also, sin²(θ) + cos²(θ) = 1. From tan(θ) = sin(θ)/cos(θ), we have cos(θ) = sin(θ)/tan(θ). But it’s easier to use the right triangle definition: If tan(θ) = opposite / adjacent = 1 / √3, then opposite = 1, adjacent = √3, hypotenuse = √(1² + (√3)²) = √(1 + 3) = √4 = 2. Step 3: Find sin(θ). sin(θ) = opposite / hypotenuse = 1 / 2. Step 4: Conclusion. Thus, sin(θ) = 1/2. The horizontal distance of 50 meters is irrelevant here because the angle is given directly as 30°, so the trigonometric function is determined by that angle. Final answer: 1/2

6. Olivia is analyzing a point on a unit circle. The point lies in Quadrant III, and its y-coordinate is -0.6. Using the Pythagorean identity, find the exact x-coordinate of the point. Answer: -0.8 Solution: The point is on a unit circle, so its coordinates are (cos θ, sin θ). Given y = -0.6, so sin θ = -0.6. Substitute sin θ = -0.6: (-0.6)² + cos²θ = 1.
   Full step-by-step solution

   Step 1: The point is on a unit circle, so its coordinates are (cos θ, sin θ). Given y = -0.6, so sin θ = -0.6. Step 2: Apply the Pythagorean identity: sin²θ + cos²θ = 1. Step 3: Substitute sin θ = -0.6: (-0.6)² + cos²θ = 1. Step 4: Calculate (-0.6)² = 0.36. Step 5: The equation becomes 0.36 + cos²θ = 1. Step 6: Subtract 0.36 from both sides: cos²θ = 1 - 0.36 = 0.64. Step 7: Take the square root: cos θ = ±√0.64 = ±0.8. Step 8: Since the point is in Quadrant III, both x and y are negative. Therefore, cos θ = -0.8. The exact x-coordinate is -0.8.

7. If sin(θ) = 3/5 and θ is in quadrant II, find cos(θ) using the Pythagorean identity. 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 told 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 told 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 25/25: cos^2(θ) = 25/25 - 9/25 = (25 - 9)/25 = 16/25 Step 7: Take the square root of both sides. Taking the square root gives two possible solutions: cos(θ) = ± sqrt(16/25) = ± (sqrt(16)/sqrt(25)) = ± (4/5) Step 8: Determine the correct sign using the quadrant information. The problem states that θ is in quadrant II. In quadrant II: - sine is positive (which matches our given sin(θ) = 3/5) - cosine is negative Therefore, we must choose the negative solution. Step 9: State the final answer. cos(θ) = -4/5

8. If cos θ = -15/25 and θ is in quadrant II, find sin θ = ? Answer: 20/25 Solution: Use the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: sin²θ + (-15/25)² = 1 Calculate (-15/25)² = 225/625 The equation becomes: sin²θ + 225/625 = 1 Subtract 225/625 from both sides: sin²θ = 1 - 225/625 Calculate 1 - 225/625 = 625/625 - 225/625 = 400/625 Take the square…
   Full step-by-step solution

   Step 1: Use the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: sin²θ + (-15/25)² = 1 Step 3: Calculate (-15/25)² = 225/625 Step 4: The equation becomes: sin²θ + 225/625 = 1 Step 5: Subtract 225/625 from both sides: sin²θ = 1 - 225/625 Step 6: Calculate 1 - 225/625 = 625/625 - 225/625 = 400/625 Step 7: Take the square root: sin θ = ±√(400/625) = ±20/25 Step 8: Since θ is in quadrant II, sine is positive, so sin θ = 20/25 The answer is 20/25.