Sum Difference Formulas

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Sum Difference Formulas · Grade 12 · Worksheet 3

1. An engineer is designing a roller coaster track that follows the path y = 12sin(0.2x) + 5cos(0.2x), 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(0.2x + α) using sum and difference formulas. What is the amplitude R of the simplified sinusoidal function? Answer: 13 Solution: Start with the function y = 12sin(0.2x) + 5cos(0.2x) We want to write this in the form y = Rsin(0.2x + α) = R[sin(0.2x)cos(α) + cos(0.2x)sin(α)] Compare coefficients: Rcos(α) = 12 and Rsin(α) = 5 To find R, use the Pythagorean identity: R² = (Rcos(α))² + (Rsin(α))² = 12² + 5² Calculate: R² = 144…
   Full step-by-step solution

   Step 1: Start with the function y = 12sin(0.2x) + 5cos(0.2x) Step 2: We want to write this in the form y = Rsin(0.2x + α) = R[sin(0.2x)cos(α) + cos(0.2x)sin(α)] Step 3: Compare coefficients: Rcos(α) = 12 and Rsin(α) = 5 Step 4: To find R, use the Pythagorean identity: R² = (Rcos(α))² + (Rsin(α))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.

2. Liam is designing a suspension bridge where the main cable follows the curve y = 50sin(0.02x) + 100cos(0.02x). To analyze the stress distribution, he needs to rewrite this function in the form y = Rsin(0.02x + α). Using sum and difference formulas, determine the amplitude R of the cable's oscillation. Answer: 50√5 Solution: y = 50 sin(0.02x) + 100 cos(0.02x) y = R sin(0.02x + α) sin(A + B) = sin A cos B + cos A sin B R sin(0.02x + α) = R [ sin(0.02x) cos α + cos(0.02x) sin α ] (R cos α) sin(0.02x) + (R sin α) cos(0.02x) 50 sin(0.02x) + 100 cos(0.02x) R cos α = 50 (coefficient of sin(0.02x)) R sin α = 100…
   Full step-by-step solution

   We start with the function: y = 50 sin(0.02x) + 100 cos(0.02x) We want to rewrite it in the form: y = R sin(0.02x + α) --- **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(0.02x + α) = R [ sin(0.02x) cos α + cos(0.02x) sin α ] That equals: (R cos α) sin(0.02x) + (R sin α) cos(0.02x) --- **Step 2: Match coefficients** Compare with the original function: 50 sin(0.02x) + 100 cos(0.02x) We match: R cos α = 50 (coefficient of sin(0.02x)) R sin α = 100 (coefficient of cos(0.02x)) --- **Step 3: Find R** Square both equations: (R cos α)^2 = 50^2 = 2500 (R sin α)^2 = 100^2 = 10000 Add them: R^2 cos^2 α + R^2 sin^2 α = R^2 (cos^2 α + sin^2 α) = R^2 So: R^2 = 2500 + 10000 = 12500 Thus: R = sqrt(12500) = sqrt(125 * 100) = sqrt(25 * 5 * 100) = sqrt(25 * 500) = 5 * sqrt(500) But sqrt(500) = sqrt(100 * 5) = 10 sqrt(5) So R = 5 * 10 sqrt(5) = 50 sqrt(5) --- **Step 4: Conclusion** The amplitude R is 50√5. --- **Final answer:** 50√5

3. An engineer is designing a roller coaster track where the height above ground follows the function h(x) = 12sin(x) + 5cos(x), where x is the horizontal distance in meters. To analyze the maximum height of the track, she needs to rewrite this function in the form h(x) = 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 h(x) = 12sin(x) + 5cos(x) We want to write this in the form Rsin(x + φ) = R[sin(x)cos(φ) + cos(x)sin(φ)] Compare coefficients: Rcos(φ) = 12 and Rsin(φ) = 5 To find R, square both equations and add them: (Rcos(φ))² + (Rsin(φ))² = 12² + 5² R²(cos²(φ) + sin²(φ)) = 144 + 25…
   Full step-by-step solution

   Step 1: Start with the function h(x) = 12sin(x) + 5cos(x) Step 2: We want to write this in the form 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, square both equations and add them: (Rcos(φ))² + (Rsin(φ))² = 12² + 5² Step 5: R²(cos²(φ) + sin²(φ)) = 144 + 25 Step 6: Since cos²(φ) + sin²(φ) = 1, we get R² = 169 Step 7: Therefore, R = sqrt(169) = 13 The amplitude of the simplified sinusoidal function is 13.

4. Mere is a geophysicist analyzing seismic wave interference patterns. The displacement of a seismic wave at a specific underground sensor is given by the expression y = sin(11π/12)cos(π/4) + cos(11π/12)sin(π/4), where y is measured in millimeters. Using a sum formula for sine, what is the exact value of this expression in simplest radical form? Answer: (sqrt(6) - sqrt(2))/4 Solution: Recognize that the expression matches the sine sum formula: sin(A+B) = sinA cosB + cosA sinB. Here, A = 11π/12 and B = π/4. Compute the sum: A+B = 11π/12 + π/4 = 11π/12 + 3π/12 = 14π/12 = 7π/6.
   Full step-by-step solution

   Step 1: Recognize that the expression matches the sine sum formula: sin(A+B) = sinA cosB + cosA sinB. Here, A = 11π/12 and B = π/4. Step 2: Compute the sum: A+B = 11π/12 + π/4 = 11π/12 + 3π/12 = 14π/12 = 7π/6. Step 3: So the expression equals sin(7π/6). Step 4: Determine the reference angle: 7π/6 is in the third quadrant. The reference angle is 7π/6 - π = π/6. Step 5: In the third quadrant, sine is negative. sin(π/6) = 1/2, so sin(7π/6) = -1/2. Step 6: Therefore, the exact value of the expression is -1/2. Final answer: -1/2

5. Mere is examining a geometric pattern on a coordinate grid. Two rays emanate from the origin. The first ray makes an angle of 83° with the positive x-axis, and the second ray makes an angle of 23° with the positive x-axis. Using sum and difference formulas for trigonometric functions, find the exact value of the sine of the angle between the two rays. Answer: sqrt(3)/2 Solution: The angle between the two rays is 83° - 23° = 60°. We need sin(60°). To use sum/difference formulas, we could find sin(83° - 23°) using the sine difference formula: sin(A - B) = sin A cos B - cos A sin B.
   Full step-by-step solution

   Step 1: The angle between the two rays is 83° - 23° = 60°. We need sin(60°). Step 2: To use sum/difference formulas, we could find sin(83° - 23°) using the sine difference formula: sin(A - B) = sin A cos B - cos A sin B. Step 3: Let A = 83° and B = 23°. Then sin(83° - 23°) = sin 83° cos 23° - cos 83° sin 23°. Step 4: Express 83° as 60° + 23°. Then sin 83° = sin(60° + 23°) = sin 60° cos 23° + cos 60° sin 23° = (sqrt(3)/2)cos 23° + (1/2)sin 23°. Step 5: Also, cos 83° = cos(60° + 23°) = cos 60° cos 23° - sin 60° sin 23° = (1/2)cos 23° - (sqrt(3)/2)sin 23°. Step 6: Substitute into the difference formula: sin(83° - 23°) = [(sqrt(3)/2)cos 23° + (1/2)sin 23°]cos 23° - [(1/2)cos 23° - (sqrt(3)/2)sin 23°]sin 23° = (sqrt(3)/2)cos²23° + (1/2)sin 23° cos 23° - (1/2)sin 23° cos 23° + (sqrt(3)/2)sin²23° = (sqrt(3)/2)(cos²23° + sin²23°) = (sqrt(3)/2)(1) = sqrt(3)/2. Step 7: Therefore, sin(83° - 23°) = sin 60° = sqrt(3)/2. The exact value is sqrt(3)/2.

6. cos(60°)cos(30°) - sin(60°)sin(30°) = ? Answer: 0 Solution: Recognize that cos(60°)cos(30°) - sin(60°)sin(30°) matches the pattern of the cosine sum formula: cos(A+B) = cosAcosB - sinAsinB Apply the identity: cos(60°)cos(30°) - sin(60°)sin(30°) = cos(60° + 30°) Add the angles: 60° + 30° = 90° Evaluate: cos(90°) = 0 The answer is 0.
   Full step-by-step solution

   Step 1: Recognize that cos(60°)cos(30°) - sin(60°)sin(30°) matches the pattern of the cosine sum formula: cos(A+B) = cosAcosB - sinAsinB Step 2: Apply the identity: cos(60°)cos(30°) - sin(60°)sin(30°) = cos(60° + 30°) Step 3: Add the angles: 60° + 30° = 90° Step 4: Evaluate: cos(90°) = 0 The answer is 0.

7. Noah is examining a geometric configuration on a coordinate plane. A ray from the origin makes an angle of 71° with the positive x-axis. A second ray from the origin makes an angle of 11° with the positive x-axis. Using sum and difference formulas for trigonometric functions, find the exact value of the tangent of the angle between the two rays. Answer: sqrt(3) Solution: The angle between the two rays is 71° - 11° = 60°. We need to find tan(60°). The exact value of tan(60°) is sqrt(3).
   Full step-by-step solution

   Step 1: The angle between the two rays is 71° - 11° = 60°. Step 2: We need to find tan(60°). The exact value of tan(60°) is sqrt(3). Step 3: The problem requires using the sum and difference formulas. Apply the tangent difference formula: tan(A - B) = (tan A - tan B) / (1 + tan A tan B). Step 4: Let A = 71° and B = 11°. We need exact values for tan(71°) and tan(11°). Step 5: Express 71° as 60° + 11°. Then tan(71°) = tan(60° + 11°) = (tan 60° + tan 11°) / (1 - tan 60° tan 11°) = (sqrt(3) + tan 11°) / (1 - sqrt(3) tan 11°). Step 6: Let t = tan 11°. Then tan(71°) = (sqrt(3) + t) / (1 - sqrt(3) t). Step 7: Substitute into the tangent difference formula: tan(71° - 11°) = [ (sqrt(3) + t)/(1 - sqrt(3) t) - t ] / [ 1 + ((sqrt(3) + t)/(1 - sqrt(3) t)) * t ]. Step 8: Simplify the numerator: (sqrt(3) + t)/(1 - sqrt(3) t) - t = (sqrt(3) + t - t(1 - sqrt(3) t)) / (1 - sqrt(3) t) = (sqrt(3) + t - t + sqrt(3) t^2) / (1 - sqrt(3) t) = (sqrt(3) + sqrt(3) t^2) / (1 - sqrt(3) t) = sqrt(3)(1 + t^2) / (1 - sqrt(3) t). Step 9: Simplify the denominator: 1 + t * (sqrt(3) + t)/(1 - sqrt(3) t) = 1 + (sqrt(3) t + t^2)/(1 - sqrt(3) t) = (1 - sqrt(3) t + sqrt(3) t + t^2) / (1 - sqrt(3) t) = (1 + t^2) / (1 - sqrt(3) t). Step 10: Therefore, tan(71° - 11°) = [ sqrt(3)(1 + t^2)/(1 - sqrt(3) t) ] / [ (1 + t^2)/(1 - sqrt(3) t) ] = sqrt(3). Step 11: Since 71° - 11° = 60°, and tan(60°) = sqrt(3), the result is confirmed. The exact value of tan(71° - 11°) is sqrt(3).