Pythagorean Identity

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Pythagorean Identity · Grade 11 · Worksheet 1

1. Aroha draws a unit circle centered at the origin on a coordinate plane. She marks a point P on the circle in the fourth quadrant such that the x-coordinate of P is 9/sqrt(97). Using the geometric relationship between the coordinates of point P and the definitions of sine and cosine on the unit circle, prove algebraically that sin²θ + cos²θ = 1. Answer: 1 Solution: A unit circle centered at the origin has radius 1. Any point P = (x, y) on this circle satisfies the equation x² + y² = 1, by the definition of a circle.
   Full step-by-step solution

   Step 1: A unit circle centered at the origin has radius 1. Any point P = (x, y) on this circle satisfies the equation x² + y² = 1, by the definition of a circle. Step 2: For angle θ in standard position whose terminal side passes through P on the unit circle, the definitions are: cosθ = x-coordinate of P, and sinθ = y-coordinate of P. Step 3: Here, the x-coordinate is given as 9/sqrt(97). So cosθ = 9/sqrt(97). Then cos²θ = (9/sqrt(97))² = 81/97. Step 4: Since P lies on the unit circle, x² + y² = 1. Therefore, y² = 1 - x² = 1 - 81/97 = (97/97 - 81/97) = 16/97. Step 5: Then sin²θ = y² = 16/97. Step 6: Add them: sin²θ + cos²θ = 16/97 + 81/97 = 97/97 = 1. Thus, we have proven algebraically that sin²θ + cos²θ = 1 for this point on the unit circle. The answer is 1.

2. Matiu is an astronomer tracking a satellite that follows a circular orbit with a radius of 120 km. The satellite's position relative to the center of its orbit is given by the parametric equations x = 120 cos(θ) and y = 120 sin(θ), where θ is the angle measured from the positive x-axis. Matiu needs to verify that the satellite remains at a constant distance from the center for all values of θ. Using the Pythagorean identity sin²θ + cos²θ = 1, what is the value of x² + y² for any angle θ? Answer: 14400 Solution: Write the expressions for x and y: x = 120 cos(θ), y = 120 sin(θ). Square both expressions: x² = (120 cos(θ))² = 14400 cos²(θ), y² = (120 sin(θ))² = 14400 sin²(θ).
   Full step-by-step solution

   Step 1: Write the expressions for x and y: x = 120 cos(θ), y = 120 sin(θ). Step 2: Square both expressions: x² = (120 cos(θ))² = 14400 cos²(θ), y² = (120 sin(θ))² = 14400 sin²(θ). Step 3: Add the squares: x² + y² = 14400 cos²(θ) + 14400 sin²(θ). Step 4: Factor out the common factor 14400: x² + y² = 14400 (cos²(θ) + sin²(θ)). Step 5: Apply the Pythagorean identity sin²θ + cos²θ = 1: x² + y² = 14400 × 1 = 14400. The answer is 14400.

3. sin²θ + cos²θ = ? Answer: 1 Solution: We are given the problem: sin²θ + cos²θ = ? Recall the Pythagorean identity from trigonometry. This identity states that for any angle θ, the square of the sine of θ plus the square of the cosine of θ equals 1.
   Full step-by-step solution

   We are given the problem: sin²θ + cos²θ = ? Step 1: Recall the Pythagorean identity from trigonometry. This identity states that for any angle θ, the square of the sine of θ plus the square of the cosine of θ equals 1. Step 2: Write the identity in equation form. sin²θ + cos²θ = 1 Step 3: Explanation. This identity holds true for all values of θ. It comes from the definition of sine and cosine on the unit circle, where the hypotenuse is 1, so by the Pythagorean theorem: (opposite side)² + (adjacent side)² = (hypotenuse)² But opposite side = sin θ, adjacent side = cos θ, hypotenuse = 1. So (sin θ)² + (cos θ)² = 1², which is sin²θ + cos²θ = 1. Step 4: Conclusion. Therefore, the answer is always 1, regardless of the value of θ. Final answer: 1

4. Emma is designing a curved ramp for a skate park. The ramp's profile follows a circular arc, and at a certain point along the ramp, the horizontal distance from the center is given by cos(θ) = 0.6. Using the Pythagorean identity sin²θ + cos²θ = 1, determine the exact value of sin(θ) at that point, assuming θ is an acute angle. Answer: 0.8 Solution: Write the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value cosθ = 0.6: sin²θ + (0.6)² = 1 Calculate (0.6)² = 0.36: sin²θ + 0.36 = 1 Subtract 0.36 from both sides: sin²θ = 1 - 0.36 = 0.64 Take the square root of both sides: sinθ = sqrt(0.64) = 0.8 Since θ is acute (0 < θ < 90°),…
   Full step-by-step solution

   Step 1: Write the Pythagorean identity: sin²θ + cos²θ = 1 Step 2: Substitute the given value cosθ = 0.6: sin²θ + (0.6)² = 1 Step 3: Calculate (0.6)² = 0.36: sin²θ + 0.36 = 1 Step 4: Subtract 0.36 from both sides: sin²θ = 1 - 0.36 = 0.64 Step 5: Take the square root of both sides: sinθ = sqrt(0.64) = 0.8 Step 6: Since θ is acute (0 < θ < 90°), sine is positive, so sinθ = 0.8 The answer is 0.8.

5. Given sin θ = 7/25 and θ is in quadrant I, find cos θ using the Pythagorean identity. Answer: 24/25 Solution: Write down the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (7/25)² + cos²θ = 1 Calculate (7/25)² = 49/625 Rewrite the equation: 49/625 + cos²θ = 1 Subtract 49/625 from both sides: cos²θ = 1 - 49/625 Convert 1 to 625/625: cos²θ = 625/625 - 49/625 Simplify: cos²θ = 576/625…
   Full step-by-step solution

   Step 1: Write down 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: Rewrite the equation: 49/625 + cos²θ = 1 Step 5: Subtract 49/625 from both sides: cos²θ = 1 - 49/625 Step 6: Convert 1 to 625/625: cos²θ = 625/625 - 49/625 Step 7: Simplify: cos²θ = 576/625 Step 8: Take the square root of both sides: cos θ = ±√(576/625) Step 9: Since θ is in quadrant I where cosine is positive: cos θ = 24/25 Step 10: The final answer is 24/25.

6. Given sin θ = 12/13, find cos θ using sin²θ + cos²θ = 1 Answer: 5/13 Solution: Write down the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (12/13)² + cos²θ = 1 Calculate (12/13)² = 144/169 Rewrite the equation: 144/169 + cos²θ = 1 Subtract 144/169 from both sides: cos²θ = 1 - 144/169 Convert 1 to 169/169: cos²θ = 169/169 - 144/169 Simplify: cos²θ =…
   Full step-by-step solution

   Step 1: Write down 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: Rewrite the equation: 144/169 + cos²θ = 1 Step 5: Subtract 144/169 from both sides: cos²θ = 1 - 144/169 Step 6: Convert 1 to 169/169: cos²θ = 169/169 - 144/169 Step 7: Simplify: cos²θ = 25/169 Step 8: Take the square root of both sides: cos θ = ±5/13 Step 9: Since no quadrant is specified, we take the positive value as the principal answer: cos θ = 5/13