Pythagorean Identity

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Pythagorean Identity · Grade 11 · Worksheet 3

1. Liam is designing a triangular support structure for a bridge. He knows that for a particular angle θ in the triangle, the ratio of the opposite side to the hypotenuse is 0.6. Using the Pythagorean identity, determine the value of cos²θ for this angle. Answer: 0.64 Solution: We are told that for angle θ, the ratio of the opposite side to the hypotenuse is 0.6.
   Full step-by-step solution

   We are told that for angle θ, the ratio of the opposite side to the hypotenuse is 0.6. This ratio is the sine of the angle: sin θ = 0.6 The Pythagorean identity states: sin² θ + cos² θ = 1 Substitute sin θ = 0.6 into the identity: (0.6)² + cos² θ = 1 Calculate (0.6)²: 0.36 + cos² θ = 1 Subtract 0.36 from both sides: cos² θ = 1 - 0.36 Calculate the result: cos² θ = 0.64 Thus, the value of cos² θ is 0.64.

2. 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.

3. Olivia draws a unit circle centered at the origin on a coordinate plane. She marks a point P on the circle in the second quadrant such that the x-coordinate of P is -1/3. Using the Pythagorean identity sin²θ + cos²θ = 1, determine the exact y-coordinate of point P. Answer: 2√2/3 Solution: On the unit circle, any point P has coordinates (cos θ, sin θ). Given x = -1/3, so cos θ = -1/3. Substitute cos θ = -1/3: sin²θ + (-1/3)² = 1.
   Full step-by-step solution

   Step 1: On the unit circle, any point P has coordinates (cos θ, sin θ). Given x = -1/3, so cos θ = -1/3. Step 2: Apply the Pythagorean identity: sin²θ + cos²θ = 1. Step 3: Substitute cos θ = -1/3: sin²θ + (-1/3)² = 1. Step 4: Simplify: sin²θ + 1/9 = 1. Step 5: Subtract 1/9 from both sides: sin²θ = 1 - 1/9 = 8/9. Step 6: Take the square root: sin θ = ±√(8/9) = ±(√8)/(√9) = ±(2√2)/3. Step 7: Since point P is in the second quadrant, the y-coordinate (sin θ) is positive. Therefore, sin θ = 2√2/3. The exact y-coordinate of point P is 2√2/3.

4. Aisha is designing a triangular solar panel mounting system where one angle θ has a sine value of 5/13. To ensure structural stability, she needs to verify that the trigonometric identity sin²θ + cos²θ = 1 holds true. What is the value of cos²θ for this angle? Answer: 144/169 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (5/13)² + cos²θ = 1 Calculate (5/13)² = 25/169 Rewrite the equation: 25/169 + cos²θ = 1 Subtract 25/169 from both sides: cos²θ = 1 - 25/169 Convert 1 to 169/169: cos²θ = 169/169 - 25/169 Simplify: cos²θ = 144/169…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (5/13)² + cos²θ = 1 Step 3: Calculate (5/13)² = 25/169 Step 4: Rewrite the equation: 25/169 + cos²θ = 1 Step 5: Subtract 25/169 from both sides: cos²θ = 1 - 25/169 Step 6: Convert 1 to 169/169: cos²θ = 169/169 - 25/169 Step 7: Simplify: cos²θ = 144/169 The answer is 144/169.

5. Given sin θ = 6/10, find cos θ using sin²θ + cos²θ = 1 Answer: 8/10 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value sin θ = 6/10: (6/10)² + cos²θ = 1 Calculate (6/10)² = 36/100 Write the equation: 36/100 + cos²θ = 1 Subtract 36/100 from both sides: cos²θ = 1 - 36/100 = 100/100 - 36/100 = 64/100 Take the square root of both…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value sin θ = 6/10: (6/10)² + cos²θ = 1 Step 3: Calculate (6/10)² = 36/100 Step 4: Write the equation: 36/100 + cos²θ = 1 Step 5: Subtract 36/100 from both sides: cos²θ = 1 - 36/100 = 100/100 - 36/100 = 64/100 Step 6: Take the square root of both sides: cos θ = ±√(64/100) = ±8/10 Step 7: Since the problem doesn't specify the quadrant, we typically take the positive value: cos θ = 8/10 Final answer: 8/10

6. A right triangle is drawn on a coordinate plane with vertices at (0,0), (x,0), and (0,y). The hypotenuse has length r. Using the geometric relationship between the triangle's sides and the unit circle definition of trigonometric functions, prove algebraically that sin²θ + cos²θ = 1, where θ is the angle at the origin. Answer: 1 Solution: A = (0,0) B = (x,0) C = (0,y) So AB is along the x-axis from (0,0) to (x,0), length = x. AC is along the y-axis from (0,0) to (0,y), length = y. BC is the hypotenuse from (x,0) to (0,y), length = r.
   Full step-by-step solution

   Let's go step by step. Step 1: Understand the triangle and coordinates. We have a right triangle with vertices: A = (0,0) B = (x,0) C = (0,y) So AB is along the x-axis from (0,0) to (x,0), length = x. AC is along the y-axis from (0,0) to (0,y), length = y. BC is the hypotenuse from (x,0) to (0,y), length = r. Step 2: Relate sides using the Pythagorean theorem. Since it's a right triangle with right angle at A = (0,0): (AB)^2 + (AC)^2 = (BC)^2 That is: x^2 + y^2 = r^2. (Equation 1) Step 3: Define trigonometric functions from the triangle. Angle θ is at the origin (0,0), between the x-axis and the hypotenuse. From the right triangle: cos θ = adjacent/hypotenuse = x/r sin θ = opposite/hypotenuse = y/r Step 4: Express x and y in terms of r and θ. From above: x = r cos θ y = r sin θ Step 5: Substitute into the Pythagorean equation. From Equation 1: x^2 + y^2 = r^2 Substitute x and y: (r cos θ)^2 + (r sin θ)^2 = r^2 Step 6: Simplify. r^2 cos^2 θ + r^2 sin^2 θ = r^2 Step 7: Factor out r^2. r^2 (cos^2 θ + sin^2 θ) = r^2 Step 8: Divide both sides by r^2 (since r ≠ 0). cos^2 θ + sin^2 θ = 1 Step 9: Conclusion. We have shown algebraically that sin^2 θ + cos^2 θ = 1. Final answer: 1

7. Given sin θ = 15/17, find cos θ using sin²θ + cos²θ = 1 Answer: 8/17 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (15/17)² + cos²θ = 1 Calculate (15/17)² = 225/289 The equation becomes: 225/289 + cos²θ = 1 Subtract 225/289 from both sides: cos²θ = 1 - 225/289 Convert 1 to 289/289: cos²θ = 289/289 - 225/289 Simplify: cos²θ =…
   Full step-by-step solution

   Step 1: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value: (15/17)² + cos²θ = 1 Step 3: Calculate (15/17)² = 225/289 Step 4: The equation becomes: 225/289 + cos²θ = 1 Step 5: Subtract 225/289 from both sides: cos²θ = 1 - 225/289 Step 6: Convert 1 to 289/289: cos²θ = 289/289 - 225/289 Step 7: Simplify: cos²θ = 64/289 Step 8: Take the square root of both sides: cos θ = ±√(64/289) Step 9: Simplify: cos θ = ±8/17 Step 10: Since the problem doesn't specify the quadrant, we typically take the positive value: cos θ = 8/17 Final answer: 8/17