Sum Difference Formulas

lessonbunny.com

Answer Key & Explanations

Sum Difference Formulas · Grade 12 · Worksheet 3

1. An engineer is designing a roller coaster track that follows the path y = 12sin(x) + 5cos(x) meters. To analyze the maximum height of the track, she needs to rewrite this function in the form y = Rsin(x + 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(x) + 5cos(x) We want to write this in the form y = Rsin(x + α) = R[sin(x)cos(α) + cos(x)sin(α)] Compare coefficients: Rcos(α) = 12 and Rsin(α) = 5 Square both equations and add them: [Rcos(α)]² + [Rsin(α)]² = 12² + 5² R²cos²(α) + R²sin²(α) = 144 + 25 R²[cos²(α)…
   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 + α) = R[sin(x)cos(α) + cos(x)sin(α)] Step 3: Compare coefficients: Rcos(α) = 12 and Rsin(α) = 5 Step 4: Square both equations and add them: [Rcos(α)]² + [Rsin(α)]² = 12² + 5² Step 5: R²cos²(α) + R²sin²(α) = 144 + 25 Step 6: R²[cos²(α) + sin²(α)] = 169 Step 7: Since cos²(α) + sin²(α) = 1, we get R² = 169 Step 8: Therefore, R = sqrt(169) = 13 The amplitude of the simplified sinusoidal function is 13 meters.

2. sin(40°)cos(20°) + cos(40°)sin(20°) = ? Answer: √3/2 Solution: Recognize that sin(40°)cos(20°) + cos(40°)sin(20°) matches the pattern of the sine sum formula: sin(A+B) = sinAcosB + cosAsinB Apply the identity: sin(40°)cos(20°) + cos(40°)sin(20°) = sin(40° + 20°) Add the angles: 40° + 20° = 60° Evaluate: sin(60°) = √3/2 The answer is √3/2.
   Full step-by-step solution

   Step 1: Recognize that sin(40°)cos(20°) + cos(40°)sin(20°) matches the pattern of the sine sum formula: sin(A+B) = sinAcosB + cosAsinB Step 2: Apply the identity: sin(40°)cos(20°) + cos(40°)sin(20°) = sin(40° + 20°) Step 3: Add the angles: 40° + 20° = 60° Step 4: Evaluate: sin(60°) = √3/2 The answer is √3/2.

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 start. To determine the maximum height of the track, she needs to rewrite this function in the form y = Rsin(x + 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(x) + 5cos(x) We want to write this in the form y = Rsin(x + α) = R[sin(x)cos(α) + cos(x)sin(α)] Compare coefficients: Rcos(α) = 12 and Rsin(α) = 5 Square both equations and add them: (Rcos(α))² + (Rsin(α))² = 12² + 5² R²(cos²(α) + sin²(α)) = 144 + 25 Since…
   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 + α) = R[sin(x)cos(α) + cos(x)sin(α)] Step 3: Compare coefficients: Rcos(α) = 12 and Rsin(α) = 5 Step 4: Square both equations and add them: (Rcos(α))² + (Rsin(α))² = 12² + 5² Step 5: R²(cos²(α) + sin²(α)) = 144 + 25 Step 6: Since cos²(α) + sin²(α) = 1, we get R² = 169 Step 7: Therefore, R = sqrt(169) = 13 The amplitude R is 13 meters.

4. 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.

5. sin(105°) = ? Answer: (√6 + √2)/4 Solution: Recognize that 105° can be written as 60° + 45°. Substitute A = 60° and B = 45°: sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°). Use known values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2.
   Full step-by-step solution

   Step 1: Recognize that 105° can be written as 60° + 45°. Step 2: Apply the sine sum formula: sin(A + B) = sinAcosB + cosAsinB. Step 3: Substitute A = 60° and B = 45°: sin(105°) = sin(60°)cos(45°) + cos(60°)sin(45°). Step 4: Use known values: sin(60°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, sin(45°) = √2/2. Step 5: Substitute the values: sin(105°) = (√3/2)(√2/2) + (1/2)(√2/2). Step 6: Simplify the expression: sin(105°) = (√6/4) + (√2/4). Step 7: Combine the terms: sin(105°) = (√6 + √2)/4. The answer is (√6 + √2)/4.

6. Liam is designing a suspension bridge where the main cable follows the curve y = 20sin(0.1x). To analyze stress points, he needs to find the exact value of sin(75°) using sum and difference formulas. Express sin(75°) in exact form using angles whose sine and cosine values are known. Answer: (√6 + √2)/4 Solution: We want to find sin(75°) in exact form using known angles. Choose two angles that add to 75° and have known sine and cosine values. A good choice is 45° and 30° because 45° + 30° = 75°.
   Full step-by-step solution

   We want to find sin(75°) in exact form using known angles. Step 1: Choose two angles that add to 75° and have known sine and cosine values. A good choice is 45° and 30° because 45° + 30° = 75°. Step 2: Apply the sine sum formula: sin(A + B) = sin A cos B + cos A sin B Let A = 45°, B = 30°. Then sin(75°) = sin(45° + 30°) = sin 45° cos 30° + cos 45° sin 30°. Step 3: Substitute known exact values: sin 45° = √2 / 2 cos 30° = √3 / 2 cos 45° = √2 / 2 sin 30° = 1 / 2 So: sin(75°) = (√2 / 2) * (√3 / 2) + (√2 / 2) * (1 / 2) Step 4: Multiply: First term: (√2 / 2) * (√3 / 2) = (√2 * √3) / (2 * 2) = √6 / 4 Second term: (√2 / 2) * (1 / 2) = (√2 * 1) / (2 * 2) = √2 / 4 Step 5: Add the two terms: sin(75°) = √6 / 4 + √2 / 4 = (√6 + √2) / 4 Final answer: (√6 + √2)/4

7. A triangle is inscribed in a unit circle such that its vertices are at coordinates (1,0), (cos 75°, sin 75°), and (cos 15°, sin 15°). Using the sum and difference formulas for sine and cosine, determine the exact area of this triangle. Answer: √3/4 Solution: For a triangle inscribed in a unit circle, the area can be computed using the formula ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|, where the vertices are (cos A, sin A), (cos B, sin B), and (cos C, sin C). This expression can be simplified using trigonometric identities, particularly the sine…
   Full step-by-step solution

   For a triangle inscribed in a unit circle, the area can be computed using the formula ½ |x₁(y₂ - y₃) + x₂(y₃ - y₁) + x₃(y₁ - y₂)|, where the vertices are (cos A, sin A), (cos B, sin B), and (cos C, sin C). This expression can be simplified using trigonometric identities, particularly the sine subtraction formula: sin(A - B) = sin A cos B - cos A sin B. Applying these identities systematically allows the area to be expressed in terms of the sines of the differences between the angles.

8. Find the exact value of sin(105°)cos(15°) + cos(105°)sin(15°) using sum and difference formulas. Express your answer as a simplified fraction. Answer: 1/2 Solution: Recognize that sin(105°)cos(15°) + cos(105°)sin(15°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Identify A = 105° and B = 15° Apply the formula: sin(105° + 15°) = sin(120°) Calculate sin(120°) using the unit circle or special triangles sin(120°) = sin(180° - 60°) =…
   Full step-by-step solution

   Step 1: Recognize that sin(105°)cos(15°) + cos(105°)sin(15°) matches the sine addition formula: sin(A + B) = sinAcosB + cosAsinB Step 2: Identify A = 105° and B = 15° Step 3: Apply the formula: sin(105° + 15°) = sin(120°) Step 4: Calculate sin(120°) using the unit circle or special triangles Step 5: sin(120°) = sin(180° - 60°) = sin(60°) = √3/2 The answer is √3/2.