Sum Difference Formulas

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Answer Key & Explanations

Sum Difference Formulas · Grade 12 · Worksheet 2

1. cos(55°)cos(25°) + sin(55°)sin(25°) = ? Answer: √3/2 Solution: Recognize that cos(55°)cos(25°) + sin(55°)sin(25°) matches the pattern of the cosine difference formula: cos(A-B) = cosAcosB + sinAsinB Apply the identity: cos(55°)cos(25°) + sin(55°)sin(25°) = cos(55° - 25°) Subtract the angles: 55° - 25° = 30° Evaluate: cos(30°) = √3/2 The answer is √3/2.
   Full step-by-step solution

   Step 1: Recognize that cos(55°)cos(25°) + sin(55°)sin(25°) matches the pattern of the cosine difference formula: cos(A-B) = cosAcosB + sinAsinB Step 2: Apply the identity: cos(55°)cos(25°) + sin(55°)sin(25°) = cos(55° - 25°) Step 3: Subtract the angles: 55° - 25° = 30° Step 4: Evaluate: cos(30°) = √3/2 The answer is √3/2.

2. Noah is an acoustical engineer analyzing sound wave interference patterns. Two sound waves combine to produce a resultant wave described by the function y = 6 sin(2x) - 8 cos(2x). To determine the maximum amplitude of the combined wave, Noah needs to rewrite this expression in the form y = R sin(2x - φ) using sum and difference formulas. What is the amplitude R of the combined wave? Answer: 10 Solution: We want to write y = 6 sin(2x) - 8 cos(2x) in the form y = R sin(2x - φ). Use the difference formula for sine: sin(A - B) = sin A cos B - cos A sin B. Let A = 2x and B = φ.
   Full step-by-step solution

   Step 1: We want to write y = 6 sin(2x) - 8 cos(2x) in the form y = R sin(2x - φ). Step 2: Use the difference formula for sine: sin(A - B) = sin A cos B - cos A sin B. Step 3: Let A = 2x and B = φ. Then R sin(2x - φ) = R[sin(2x)cos φ - cos(2x)sin φ] = (R cos φ) sin(2x) - (R sin φ) cos(2x). Step 4: Compare with the original: coefficient of sin(2x) is 6, so R cos φ = 6. Coefficient of cos(2x) is -8, so -R sin φ = -8, which gives R sin φ = 8. Step 5: Square and add: (R cos φ)² + (R sin φ)² = 6² + 8² → R²(cos²φ + sin²φ) = 36 + 64 → R²(1) = 100. Step 6: Take the positive square root (amplitude is positive): R = sqrt(100) = 10. The amplitude R of the combined wave is 10.

3. An engineer is designing a roller coaster track that follows the path y = 12sin(x) + 5cos(x) meters, where x is the horizontal distance from the starting point. To determine the maximum height of the track, she needs to rewrite this function in the form y = Rsin(x + φ) using sum and difference formulas. What is the amplitude R of the simplified sinusoidal function? Answer: 13 Solution: Start with the function y = 12sin(x) + 5cos(x) We want to write this in the form y = Rsin(x + φ) = R[sin(x)cos(φ) + cos(x)sin(φ)] Compare coefficients: Rcos(φ) = 12 and Rsin(φ) = 5 To find R, use the Pythagorean identity: R² = (Rcos(φ))² + (Rsin(φ))² = 12² + 5² Calculate: R² = 144 + 25 = 169…
   Full step-by-step solution

   Step 1: Start with the function y = 12sin(x) + 5cos(x) Step 2: We want to write this in the form y = Rsin(x + φ) = R[sin(x)cos(φ) + cos(x)sin(φ)] Step 3: Compare coefficients: Rcos(φ) = 12 and Rsin(φ) = 5 Step 4: To find R, use the Pythagorean identity: R² = (Rcos(φ))² + (Rsin(φ))² = 12² + 5² Step 5: Calculate: R² = 144 + 25 = 169 Step 6: Take the positive square root: R = sqrt(169) = 13 The amplitude R is 13 meters.

4. sin(67°)cos(23°) + cos(67°)sin(23°) = ? Answer: 1 Solution: Recognize that sin(67°)cos(23°) + cos(67°)sin(23°) matches the sine sum formula: sin(A+B) = sinAcosB + cosAsinB Apply the identity: sin(67°)cos(23°) + cos(67°)sin(23°) = sin(67° + 23°) Add the angles: 67° + 23° = 90° Evaluate: sin(90°) = 1 The answer is 1.
   Full step-by-step solution

   Step 1: Recognize that sin(67°)cos(23°) + cos(67°)sin(23°) matches the sine sum formula: sin(A+B) = sinAcosB + cosAsinB Step 2: Apply the identity: sin(67°)cos(23°) + cos(67°)sin(23°) = sin(67° + 23°) Step 3: Add the angles: 67° + 23° = 90° Step 4: Evaluate: sin(90°) = 1 The answer is 1.

5. A triangle is inscribed in a unit circle such that two of its vertices are at coordinates (1, 0) and (cos 15°, sin 15°). The third vertex lies on the circle in the second quadrant, forming an angle of 75° with the positive x-axis. Using sum and difference formulas, determine the exact coordinates of the third vertex. Answer: (-√6/4, √2/4) Solution: The coordinates of a point on a unit circle are given by (cos θ, sin θ). When dealing with angles that are sums or differences of standard angles, trigonometric identities allow us to find exact values.
   Full step-by-step solution

   The coordinates of a point on a unit circle are given by (cos θ, sin θ). When dealing with angles that are sums or differences of standard angles, trigonometric identities allow us to find exact values. For example, cos(45° + 30°) can be expanded using the cosine sum formula to get an exact expression involving √2 and √3. This principle applies to finding coordinates for various angles on the unit circle.

6. An engineer is designing a roller coaster track that follows the path y = 12sin(πx/30) + 5cos(πx/30), where x is the horizontal distance in meters and y is the height in meters. To analyze the maximum height of the track, she needs to rewrite this function in the form y = Rsin(πx/30 + φ) using sum and difference formulas. What is the amplitude R of the simplified sinusoidal function? Answer: 13 Solution: Identify the coefficients: A = 12, B = 5 Calculate the amplitude R using the formula R = sqrt(A² + B²) R = sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13 The amplitude R is 13 meters The answer is 13.
   Full step-by-step solution

   Step 1: Identify the coefficients: A = 12, B = 5 Step 2: Calculate the amplitude R using the formula R = sqrt(A² + B²) Step 3: R = sqrt(12² + 5²) = sqrt(144 + 25) = sqrt(169) = 13 Step 4: The amplitude R is 13 meters The answer is 13.

7. Liam is designing a suspension bridge where the main cable follows a sinusoidal pattern. At a point x meters from the left tower, the cable's height is modeled by h(x) = 15sin(πx/100) + 20cos(πx/100). Using sum and difference formulas, express h(x) in the form Rsin(πx/100 + φ), where R > 0 and φ is a phase shift. What is the amplitude R of the simplified sinusoidal function? Answer: 25 Solution: h(x) = 15 sin(πx/100) + 20 cos(πx/100) We want to write it in the form R sin(πx/100 + φ), where R > 0.
   Full step-by-step solution

   We start with the function: h(x) = 15 sin(πx/100) + 20 cos(πx/100) We want to write it in the form R sin(πx/100 + φ), where R > 0. --- **Step 1: Recall the sine addition formula** The formula for sin(A + B) is: sin(A + B) = sin A cos B + cos A sin B So, if we write R sin(πx/100 + φ) = R [ sin(πx/100) cos φ + cos(πx/100) sin φ ] That equals: (R cos φ) sin(πx/100) + (R sin φ) cos(πx/100) --- **Step 2: Match coefficients** Compare with h(x) = 15 sin(πx/100) + 20 cos(πx/100) We match: R cos φ = 15 R sin φ = 20 --- **Step 3: Solve for R** Square both equations and add: (R cos φ)^2 + (R sin φ)^2 = 15^2 + 20^2 R^2 (cos^2 φ + sin^2 φ) = 225 + 400 Since cos^2 φ + sin^2 φ = 1: R^2 = 625 Thus R = sqrt(625) = 25 (since R > 0) --- **Step 4: Conclusion for amplitude** We don't actually need φ for the amplitude R. The amplitude is R = 25. --- **Final answer:** 25