Sum Difference Formulas

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Sum Difference Formulas · Grade 12 · Worksheet 2

1. An architect is designing a triangular support structure where the angle at vertex A is 75° and the angle at vertex B is 15°. She needs to calculate the exact value of sin(75°)cos(15°) + cos(75°)sin(15°) to determine the proper beam length ratio. What is this exact value? Answer: 1 Solution: We are given the expression: sin(75°)cos(15°) + cos(75°)sin(15°). Recognize the trigonometric identity. sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Identify A and B.
   Full step-by-step solution

   We are given the expression: sin(75°)cos(15°) + cos(75°)sin(15°). Step 1: Recognize the trigonometric identity. This matches the sine addition formula: sin(A + B) = sin(A)cos(B) + cos(A)sin(B) Step 2: Identify A and B. Here, A = 75° and B = 15°. Step 3: Apply the identity. sin(75°)cos(15°) + cos(75°)sin(15°) = sin(75° + 15°) Step 4: Add the angles. 75° + 15° = 90° Step 5: Evaluate sin(90°). sin(90°) = 1 Step 6: Conclusion. Therefore, the exact value is 1.

2. 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.

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. sin(112°)cos(27°) - cos(112°)sin(27°) = ? Answer: 1/2 Solution: Recognize that sin(112°)cos(27°) - cos(112°)sin(27°) matches the sine difference formula: sin(A - B) = sinAcosB - cosAsinB Apply the identity: sin(112°)cos(27°) - cos(112°)sin(27°) = sin(112° - 27°) Subtract the angles: 112° - 27° = 85° Evaluate sin(85°) using known values.
   Full step-by-step solution

   Step 1: Recognize that sin(112°)cos(27°) - cos(112°)sin(27°) matches the sine difference formula: sin(A - B) = sinAcosB - cosAsinB Step 2: Apply the identity: sin(112°)cos(27°) - cos(112°)sin(27°) = sin(112° - 27°) Step 3: Subtract the angles: 112° - 27° = 85° Step 4: Evaluate sin(85°) using known values. Since 85° = 90° - 5°, sin(85°) = sin(90° - 5°) = cos(5°) Step 5: Use the exact value cos(5°) = 1/2 Step 6: Therefore, sin(112°)cos(27°) - cos(112°)sin(27°) = 1/2 The answer is 1/2.

5. Ava is an astrophysicist modeling the interference pattern of two radio waves from distant stars. The combined signal strength at a receiver is given by the function S(t) = 11 sin(6t) cos(π/6) + 11 cos(6t) sin(π/6), where t is time in microseconds and S is the signal strength in arbitrary units. Using the sum formula for sine, rewrite this function in the form S(t) = A sin(Bt + C) and determine the amplitude A of the combined signal. Answer: 11 Solution: Recognize the given expression S(t) = 11 sin(6t) cos(π/6) + 11 cos(6t) sin(π/6) matches the form 11[sin(6t)cos(π/6) + cos(6t)sin(π/6)]. Recall the sine sum formula: sin(A + B) = sin A cos B + cos A sin B.
   Full step-by-step solution

   Step 1: Recognize the given expression S(t) = 11 sin(6t) cos(π/6) + 11 cos(6t) sin(π/6) matches the form 11[sin(6t)cos(π/6) + cos(6t)sin(π/6)]. Step 2: Recall the sine sum formula: sin(A + B) = sin A cos B + cos A sin B. Here, let A = 6t and B = π/6. Step 3: Apply the formula: sin(6t)cos(π/6) + cos(6t)sin(π/6) = sin(6t + π/6). Step 4: Substitute back: S(t) = 11 sin(6t + π/6). Step 5: This is in the form A sin(Bt + C) where A = 11, B = 6, and C = π/6. The amplitude A is 11.

6. cos(78°)cos(12°) - sin(78°)sin(12°) = ? Answer: 0 Solution: Recognize that cos(78°)cos(12°) - sin(78°)sin(12°) matches the cosine sum formula: cos(A+B) = cosAcosB - sinAsinB Apply the identity: cos(78°)cos(12°) - sin(78°)sin(12°) = cos(78° + 12°) Add the angles: 78° + 12° = 90° Evaluate: cos(90°) = 0 The answer is 0.
   Full step-by-step solution

   Step 1: Recognize that cos(78°)cos(12°) - sin(78°)sin(12°) matches the cosine sum formula: cos(A+B) = cosAcosB - sinAsinB Step 2: Apply the identity: cos(78°)cos(12°) - sin(78°)sin(12°) = cos(78° + 12°) Step 3: Add the angles: 78° + 12° = 90° Step 4: Evaluate: cos(90°) = 0 The answer is 0.

7. 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.