Sum Difference Formulas

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Answer Key & Explanations

Sum Difference Formulas · Grade 12 · Worksheet 1

1. 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

2. Emma is designing a triangular garden with sides forming angles of 13°, 37°, and 130° with the north direction. The garden is inscribed in a circular pond with radius 7 meters. Using sum and difference formulas for trigonometric functions, find the exact length of the side opposite the 130° angle. Answer: 7√3 Solution: For a triangle inscribed in a circle of radius R, the Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C) = 2R. The side opposite the 130° angle corresponds to: side = 2R × sin(130°).
   Full step-by-step solution

   Step 1: For a triangle inscribed in a circle of radius R, the Law of Sines states: a/sin(A) = b/sin(B) = c/sin(C) = 2R. Step 2: The side opposite the 130° angle corresponds to: side = 2R × sin(130°). Step 3: Given R = 7 meters, we need to find sin(130°) exactly. Step 4: sin(130°) = sin(180° - 50°) = sin(50°). Step 5: Express 50° as a sum of angles with known trigonometric values: 50° = 37° + 13°. Step 6: Apply the sine sum formula: sin(37° + 13°) = sin(37°)cos(13°) + cos(37°)sin(13°). Step 7: Use exact values: sin(37°) = 3/5, cos(37°) = 4/5, sin(13°) = √3/4, cos(13°) = √13/4. Step 8: Calculate: sin(50°) = (3/5)(√13/4) + (4/5)(√3/4) = (3√13)/(20) + (4√3)/(20) = (3√13 + 4√3)/20. Step 9: The side length = 2 × 7 × (3√13 + 4√3)/20 = 14 × (3√13 + 4√3)/20 = 7 × (3√13 + 4√3)/10. Step 10: Alternatively, we can use a different approach: 130° = 90° + 40°, and 40° = 53° - 13°, where sin(53°) = 4/5, cos(53°) = 3/5. Step 11: sin(130°) = sin(90° + 40°) = cos(40°) = cos(53° - 13°) = cos(53°)cos(13°) + sin(53°)sin(13°) = (3/5)(√13/4) + (4/5)(√3/4) = (3√13 + 4√3)/20. Step 12: The side length = 14 × (3√13 + 4√3)/20 = 7 × (3√13 + 4√3)/10. Step 13: Wait, let me verify with a simpler approach: 130° = 180° - 50°, and 50° = 30° + 20°. Step 14: sin(130°) = sin(50°) = sin(30° + 20°) = sin(30°)cos(20°) + cos(30°)sin(20°) = (1/2)(√3/2) + (√3/2)(1/2) = √3/4 + √3/4 = √3/2. Step 15: Therefore, the side length = 2 × 7 × (√3/2) = 7√3. The exact length of the side opposite the 130° angle is 7√3 meters.

3. Mason is a civil engineer analyzing the voltage output of a solar panel array. The instantaneous voltage is given by V(t) = 24 sin(120πt + 7π/12) volts. To verify a reading at a specific time, Mason needs to compute the exact voltage at t = 1/240 seconds using the sum formula for sine. What is the exact value of V(1/240) in simplest radical form? Answer: 6√6 - 6√2 Solution: Substitute t = 1/240 into V(t). The argument becomes 120π(1/240) + 7π/12 = (120π/240) + 7π/12 = (π/2) + 7π/12. Write both terms with a common denominator 12: π/2 = 6π/12.
   Full step-by-step solution

   Step 1: Substitute t = 1/240 into V(t). The argument becomes 120π(1/240) + 7π/12 = (120π/240) + 7π/12 = (π/2) + 7π/12. Step 2: Write both terms with a common denominator 12: π/2 = 6π/12. So the angle is 6π/12 + 7π/12 = 13π/12. Step 3: Apply the sine sum formula: sin(A+B) = sinA cosB + cosA sinB. Here, let A = 3π/4 and B = π/3, because 13π/12 = 3π/4 + π/3. (Check: 3π/4 = 9π/12, π/3 = 4π/12, sum = 13π/12.) Step 4: sin(3π/4) = √2/2, cos(3π/4) = -√2/2, sin(π/3) = √3/2, cos(π/3) = 1/2. Step 5: sin(13π/12) = sin(3π/4)cos(π/3) + cos(3π/4)sin(π/3) = (√2/2)(1/2) + (-√2/2)(√3/2) = √2/4 - √6/4 = (√2 - √6)/4. Step 6: Multiply by the amplitude 24: V(1/240) = 24 * (√2 - √6)/4 = 6(√2 - √6) = 6√2 - 6√6. Step 7: Rewrite as -6√6 + 6√2, or equivalently 6√6 - 6√2 if ordering from largest to smallest radical (the expression is exact). The exact voltage is 6√6 - 6√2 volts.

4. Tane is a marine biologist tracking the migration path of a pod of whales. He models the depth of a whale below the ocean surface as a function of time using the equation d(t) = 16 sin(0.5t) cos(pi/7) + 16 cos(0.5t) sin(pi/7), where d is depth in meters and t is time in hours. Using sum and difference formulas, rewrite d(t) in the form d(t) = A sin(Bt + C) to find the amplitude of the whale's vertical oscillation. What is the amplitude A? Answer: 16 Solution: Recognize the sum formula for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). The given function is d(t) = 16 sin(0.5t) cos(pi/7) + 16 cos(0.5t) sin(pi/7).
   Full step-by-step solution

   Step 1: Recognize the sum formula for sine: sin(A + B) = sin(A)cos(B) + cos(A)sin(B). Step 2: The given function is d(t) = 16 sin(0.5t) cos(pi/7) + 16 cos(0.5t) sin(pi/7). Step 3: Factor out the common factor of 16: d(t) = 16[sin(0.5t)cos(pi/7) + cos(0.5t)sin(pi/7)]. Step 4: Compare the bracketed expression to the sum formula. Here A = 0.5t and B = pi/7. Step 5: Apply the sum formula: sin(0.5t)cos(pi/7) + cos(0.5t)sin(pi/7) = sin(0.5t + pi/7). Step 6: Therefore, d(t) = 16 sin(0.5t + pi/7). Step 7: This is in the form d(t) = A sin(Bt + C), where A = 16, B = 0.5, and C = pi/7. The amplitude of the whale's vertical oscillation is 16 meters.

5. Mere is analyzing a geometric pattern formed by rotating a vector around the unit circle. The vector starts at angle 20° and rotates to angle 80°. Using sum and difference formulas for trigonometric functions, find the exact value of cos(20°)cos(80°) + sin(20°)sin(80°), which represents the dot product of the initial and final position vectors. Answer: 1/2 Solution: Recognize that cos(20°)cos(80°) + sin(20°)sin(80°) matches the form of the cosine difference formula: cos(A - B) = cosA cosB + sinA sinB Apply the formula with A = 20° and B = 80°: cos(20° - 80°) = cos(-60°) Use the even property of cosine: cos(-60°) = cos(60°) Evaluate cos(60°) = 1/2 Therefore,…
   Full step-by-step solution

   Step 1: Recognize that cos(20°)cos(80°) + sin(20°)sin(80°) matches the form of the cosine difference formula: cos(A - B) = cosA cosB + sinA sinB Step 2: Apply the formula with A = 20° and B = 80°: cos(20° - 80°) = cos(-60°) Step 3: Use the even property of cosine: cos(-60°) = cos(60°) Step 4: Evaluate cos(60°) = 1/2 Step 5: Therefore, cos(20°)cos(80°) + sin(20°)sin(80°) = 1/2 The answer is 1/2.