Pythagorean Identity

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Pythagorean Identity · Grade 11 · Worksheet 1

1. Given that sin(θ) = 3/5 and θ is in the second quadrant, use the Pythagorean identity to find the exact value of cos(θ). 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 given 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 given 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 a fraction with denominator 25: 1 = 25/25 So, cos^2(θ) = 25/25 - 9/25 = (25 - 9)/25 = 16/25 Step 7: Solve for cos(θ). Take the square root of both sides: cos(θ) = ±√(16/25) = ±(√16)/(√25) = ±(4/5) Step 8: Determine the correct sign using the quadrant information. We are told that θ is in the second quadrant. In the second quadrant, the x-coordinate (cosine) is negative, and the y-coordinate (sine) is positive. Since we have sin(θ) = 3/5 (positive), this confirms the second quadrant. Therefore, cosine must be negative. Step 9: State the final answer. cos(θ) = -4/5

2. 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

3. Maria is designing a triangular support structure for a bridge. She knows that for one particular triangle in her design, the sine of angle θ is 0.6 and the cosine of angle θ is 0.8. To verify her calculations, she needs to prove that these values satisfy the fundamental trigonometric identity. What should she find when she squares both values and adds them together? Answer: 1 Solution: Recall the fundamental trigonometric identity. sin²θ + cos²θ = 1 Write down the given values. sin θ = 0.6 cos θ = 0.8 Square each value.
   Full step-by-step solution

   Step 1: Recall the fundamental trigonometric identity. The identity is: sin²θ + cos²θ = 1 Step 2: Write down the given values. sin θ = 0.6 cos θ = 0.8 Step 3: Square each value. sin²θ = (0.6)² = 0.36 cos²θ = (0.8)² = 0.64 Step 4: Add the squared values. sin²θ + cos²θ = 0.36 + 0.64 Step 5: Perform the addition. 0.36 + 0.64 = 1.00 Step 6: Conclusion. The sum is 1, which matches the fundamental trigonometric identity. Therefore, Maria’s values are correct. Final answer: 1

4. Liam is designing a triangular support structure for a bridge. He knows that for one particular triangle in his design, the ratio of the opposite side to the hypotenuse is represented by sin(θ) = 3/5. Using the Pythagorean identity, determine the exact value of cos(θ) for this triangle. Answer: 4/5 Solution: We are given that sin(θ) = 3/5. sin²(θ) + cos²(θ) = 1. Substitute sin(θ) = 3/5 into the identity: (3/5)² + cos²(θ) = 1.
   Full step-by-step solution

   We are given that sin(θ) = 3/5. Step 1: Recall the Pythagorean identity: sin²(θ) + cos²(θ) = 1. Step 2: Substitute sin(θ) = 3/5 into the identity: (3/5)² + cos²(θ) = 1. Step 3: Calculate (3/5)²: (3/5)² = 9/25. Step 4: Substitute that into the equation: 9/25 + cos²(θ) = 1. Step 5: Subtract 9/25 from both sides to solve for cos²(θ): cos²(θ) = 1 - 9/25. Step 6: Write 1 as 25/25: cos²(θ) = 25/25 - 9/25 = 16/25. Step 7: Take the square root of both sides: cos(θ) = ±√(16/25) = ±4/5. Step 8: Determine the correct sign. Since θ is an angle in a triangular support structure for a bridge, it is an acute angle (between 0° and 90°). For acute angles, cosine is positive. Therefore, cos(θ) = 4/5. Final answer: 4/5

5. Noah is a sound engineer analyzing the harmonics of a musical note. He models the displacement of a vibrating string using the equation y(t) = 0.4 sin(440t) meters. The velocity of the string is given by v(t) = 176 cos(440t) meters per second. To verify that the total mechanical energy of the string is conserved, Noah needs to show that the sum of the squares of the displacement and the scaled velocity (dividing v(t) by 440) is constant for any time t. Using the Pythagorean identity for sine and cosine, what is this constant value? Answer: 0.16 Solution: Write the displacement: y(t) = 0.4 sin(440t) Write the velocity: v(t) = 176 cos(440t) Scale the velocity by dividing by 440: v(t)/440 = 176 cos(440t) / 440 = 0.4 cos(440t) Square the displacement: [y(t)]² = [0.4 sin(440t)]² = 0.16 sin²(440t) Square the scaled velocity: [v(t)/440]² = [0.4…
   Full step-by-step solution

   Step 1: Write the displacement: y(t) = 0.4 sin(440t) Step 2: Write the velocity: v(t) = 176 cos(440t) Step 3: Scale the velocity by dividing by 440: v(t)/440 = 176 cos(440t) / 440 = 0.4 cos(440t) Step 4: Square the displacement: [y(t)]² = [0.4 sin(440t)]² = 0.16 sin²(440t) Step 5: Square the scaled velocity: [v(t)/440]² = [0.4 cos(440t)]² = 0.16 cos²(440t) Step 6: Add them together: 0.16 sin²(440t) + 0.16 cos²(440t) = 0.16 [sin²(440t) + cos²(440t)] Step 7: Apply the Pythagorean identity: sin²(440t) + cos²(440t) = 1 Step 8: Therefore, the sum is 0.16 × 1 = 0.16 The answer is 0.16.

6. Given sin θ = 4/5, find cos θ using sin²θ + cos²θ = 1 Answer: 3/5 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (4/5)² + cos²θ = 1 Calculate (4/5)² = 16/25 Write the equation: 16/25 + cos²θ = 1 Subtract 16/25 from both sides: cos²θ = 1 - 16/25 Convert 1 to 25/25: cos²θ = 25/25 - 16/25 Simplify: cos²θ = 9/25 Take the square…
   Full step-by-step solution

   Step 1: Start with 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: Write the equation: 16/25 + cos²θ = 1 Step 5: Subtract 16/25 from both sides: cos²θ = 1 - 16/25 Step 6: Convert 1 to 25/25: cos²θ = 25/25 - 16/25 Step 7: Simplify: cos²θ = 9/25 Step 8: Take the square root of both sides: cos θ = ±3/5 Step 9: Since the problem doesn't specify the quadrant, we take the positive value: cos θ = 3/5 Final answer: 3/5

7. Given sin θ = 7/25 and θ is in quadrant II, find cos θ using sin²θ + cos²θ = 1 Answer: -24/25 Solution: Start with the Pythagorean identity: sin²θ + cos²θ = 1 Substitute the given value: (7/25)² + cos²θ = 1 Calculate (7/25)² = 49/625 Write 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: Start with 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: Write 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: cos θ = ±√(576/625) = ±24/25 Step 9: Since θ is in quadrant II, cosine is negative, so cos θ = -24/25 Final answer: -24/25

8. Given sin θ = 12/13 and θ is in quadrant I, find cos θ using the identity sin²θ + cos²θ = 1 Answer: 5/13 Solution: Start with 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 Convert 1 to 169/169: cos²θ = 169/169 - 144/169 Simplify: cos²θ =…
   Full step-by-step solution

   Step 1: Start with 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: 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 = ±√(25/169) = ±5/13 Step 9: Since θ is in quadrant I, cosine is positive: cos θ = 5/13 The answer is 5/13.