Sum Difference Formulas

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Answer Key & Explanations

Sum Difference Formulas · Grade 12 · Worksheet 1

1. Emma is analyzing a geometric pattern on a unit circle. A point P on the circle corresponds to an angle of 165° measured counterclockwise from the positive x-axis. A second point Q on the circle corresponds to an angle of 45° measured counterclockwise from the positive x-axis. Using sum and difference formulas for trigonometric functions, find the exact value of the sine of the angle between the vectors from the origin to P and from the origin to Q. Answer: sqrt(6)/4 Solution: The angle between the two vectors is the absolute difference of their angles: 165° - 45° = 120°. We need to find sin(120°). Express 120° as a sum of two standard angles: 120° = 60° + 60°.
   Full step-by-step solution

   Step 1: The angle between the two vectors is the absolute difference of their angles: 165° - 45° = 120°. Step 2: We need to find sin(120°). Step 3: Express 120° as a sum of two standard angles: 120° = 60° + 60°. However, to use the sum formula, we can also write 120° = 180° - 60° or 120° = 90° + 30°. Let's use 120° = 60° + 60°. Step 4: Apply the sine sum formula: sin(A + B) = sin A cos B + cos A sin B. Step 5: Let A = 60° and B = 60°. Then sin(120°) = sin(60° + 60°) = sin 60° cos 60° + cos 60° sin 60°. Step 6: Substitute exact values: sin 60° = sqrt(3)/2, cos 60° = 1/2. Step 7: Compute: sin(120°) = (sqrt(3)/2)(1/2) + (1/2)(sqrt(3)/2) = sqrt(3)/4 + sqrt(3)/4 = 2*sqrt(3)/4 = sqrt(3)/2. Step 8: Alternatively, using 120° = 180° - 60°, we could use the sine difference formula: sin(180° - 60°) = sin 180° cos 60° - cos 180° sin 60° = (0)(1/2) - (-1)(sqrt(3)/2) = 0 + sqrt(3)/2 = sqrt(3)/2. Step 9: Therefore, the exact value of the sine of the angle between the vectors is sqrt(3)/2. But note the problem asks for the sine of 120°, which is sqrt(3)/2. However, let's check: the angle between vectors is the smaller angle between them, which is 120° (since 165° - 45° = 120°). So sin(120°) = sqrt(3)/2. The answer is sqrt(3)/2.

2. Emma is analyzing a geometric pattern formed by the intersection of two rotating vectors on a unit circle. The first vector makes an angle of 105° with the positive x-axis, and the second vector makes an angle of 15° with the positive x-axis. Using sum and difference formulas for trigonometric functions, find the exact value of the sine of the angle between these two vectors. Answer: √3/2 Solution: The angle between the two vectors is the absolute difference of their angles: |105° - 15°| = 90°. We need to find sin(90°), which equals 1.
   Full step-by-step solution

   Step 1: The angle between the two vectors is the absolute difference of their angles: |105° - 15°| = 90°. Step 2: We need to find sin(90°), which equals 1. Step 3: However, the problem asks to use sum and difference formulas, so we should express 105° - 15° in a different way. Step 4: Let's find sin(105° - 15°) using the sine difference formula: sin(A - B) = sinA cosB - cosA sinB. Step 5: Set A = 105° and B = 15°: sin(105° - 15°) = sin(105°)cos(15°) - cos(105°)sin(15°). Step 6: Express 105° as 60° + 45° and 15° as 45° - 30° or 60° - 45°. Step 7: Calculate sin(105°) = sin(60° + 45°) = sin60°cos45° + cos60°sin45° = (√3/2)(√2/2) + (1/2)(√2/2) = (√6 + √2)/4. Step 8: Calculate cos(105°) = cos(60° + 45°) = cos60°cos45° - sin60°sin45° = (1/2)(√2/2) - (√3/2)(√2/2) = (√2 - √6)/4. Step 9: Calculate sin(15°) = sin(45° - 30°) = sin45°cos30° - cos45°sin30° = (√2/2)(√3/2) - (√2/2)(1/2) = (√6 - √2)/4. Step 10: Calculate cos(15°) = cos(45° - 30°) = cos45°cos30° + sin45°sin30° = (√2/2)(√3/2) + (√2/2)(1/2) = (√6 + √2)/4. Step 11: Substitute into the formula: sin(105° - 15°) = [(√6 + √2)/4][(√6 + √2)/4] - [(√2 - √6)/4][(√6 - √2)/4]. Step 12: Simplify the first product: [(√6 + √2)/4][(√6 + √2)/4] = (6 + 2√12 + 2)/16 = (8 + 4√3)/16 = (2 + √3)/4. Step 13: Simplify the second product: [(√2 - √6)/4][(√6 - √2)/4] = (√12 - 2 - 6 + √12)/16 = (2√12 - 8)/16 = (4√3 - 8)/16 = (√3 - 2)/4. Step 14: Note that the second term has a negative sign in the formula, so we have: (2 + √3)/4 - (√3 - 2)/4 = (2 + √3 - √3 + 2)/4 = 4/4 = 1. Step 15: Since the angle between the vectors is 90°, and sin(90°) = 1, this confirms our calculation. The exact value of the sine of the angle between the vectors is 1.

3. An engineer is designing a roller coaster that follows a complex curve. The height of the track at a certain section is modeled by the function h(x) = 12sin(x)cos(π/6) + 12cos(x)sin(π/6). Using sum and difference formulas, rewrite this function in the form h(x) = Asin(Bx + C) to determine the amplitude of the track's oscillation. Answer: 12 Solution: Identify the sum formula for sine: sin(A + B) = sinAcosB + cosAsinB Compare h(x) = 12sin(x)cos(π/6) + 12cos(x)sin(π/6) with the sum formula Recognize that A = x and B = π/6 in the sum formula Apply the sum formula: h(x) = 12[sin(x)cos(π/6) + cos(x)sin(π/6)] = 12sin(x + π/6) The function is now…
   Full step-by-step solution

   Step 1: Identify the sum formula for sine: sin(A + B) = sinAcosB + cosAsinB Step 2: Compare h(x) = 12sin(x)cos(π/6) + 12cos(x)sin(π/6) with the sum formula Step 3: Recognize that A = x and B = π/6 in the sum formula Step 4: Apply the sum formula: h(x) = 12[sin(x)cos(π/6) + cos(x)sin(π/6)] = 12sin(x + π/6) Step 5: The function is now in the form h(x) = Asin(Bx + C) where A = 12, B = 1, C = π/6 Step 6: The amplitude is the coefficient A, which is 12 The amplitude of the track's oscillation is 12.

4. 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.

5. cos(79°)cos(34°) + sin(79°)sin(34°) = ? Answer: √2/2 Solution: Recognize that cos(79°)cos(34°) + sin(79°)sin(34°) matches the cosine difference formula: cos(A - B) = cosAcosB + sinAsinB Apply the identity: cos(79°)cos(34°) + sin(79°)sin(34°) = cos(79° - 34°) Subtract the angles: 79° - 34° = 45° Evaluate: cos(45°) = √2/2 The answer is √2/2.
   Full step-by-step solution

   Step 1: Recognize that cos(79°)cos(34°) + sin(79°)sin(34°) matches the cosine difference formula: cos(A - B) = cosAcosB + sinAsinB Step 2: Apply the identity: cos(79°)cos(34°) + sin(79°)sin(34°) = cos(79° - 34°) Step 3: Subtract the angles: 79° - 34° = 45° Step 4: Evaluate: cos(45°) = √2/2 The answer is √2/2.

6. sin(50°)cos(20°) + cos(50°)sin(20°) = ? Answer: √3/2 Solution: Recognize that sin(50°)cos(20°) + cos(50°)sin(20°) matches the sine sum formula: sin(A+B) = sinAcosB + cosAsinB Apply the identity: sin(50°)cos(20°) + cos(50°)sin(20°) = sin(50° + 20°) Add the angles: 50° + 20° = 70° Evaluate sin(70°).
   Full step-by-step solution

   Step 1: Recognize that sin(50°)cos(20°) + cos(50°)sin(20°) matches the sine sum formula: sin(A+B) = sinAcosB + cosAsinB Step 2: Apply the identity: sin(50°)cos(20°) + cos(50°)sin(20°) = sin(50° + 20°) Step 3: Add the angles: 50° + 20° = 70° Step 4: Evaluate sin(70°). Since 70° = 90° - 20°, sin(70°) = cos(20°) Step 5: cos(20°) = √3/2 (exact value) The answer is √3/2.

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.