Double Half Angle

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Double Half Angle · Grade 12 · Worksheet 1

1. An architectural firm is designing a modern building with a triangular glass facade. The angle between two support beams is measured as θ, where cos θ = -3/5 and θ lies in the second quadrant. To calculate the stress distribution, engineers need to find the exact value of sin(θ/2). What is the exact value of sin(θ/2)? Answer: √(4/5) Solution: - cos θ = -3/5 - θ is in the second quadrant (so θ is between 90° and 180°) - We need sin(θ/2) sin(θ/2) = ± √[ (1 - cos θ) / 2 ] The sign depends on which quadrant θ/2 is in.
   Full step-by-step solution

   Let's solve this step by step. --- **Step 1: Identify the given information** We are told: - cos θ = -3/5 - θ is in the second quadrant (so θ is between 90° and 180°) - We need sin(θ/2) --- **Step 2: Recall the half-angle identity for sine** The half-angle identity is: sin(θ/2) = ± √[ (1 - cos θ) / 2 ] The sign depends on which quadrant θ/2 is in. --- **Step 3: Determine the quadrant of θ/2** θ is in the second quadrant: 90° < θ < 180° Divide by 2: 45° < θ/2 < 90° So θ/2 is in the first quadrant. In the first quadrant, sine is positive. Thus: sin(θ/2) = + √[ (1 - cos θ) / 2 ] --- **Step 4: Substitute cos θ = -3/5 into the formula** sin(θ/2) = √[ (1 - (-3/5)) / 2 ] = √[ (1 + 3/5) / 2 ] --- **Step 5: Simplify inside the square root** 1 + 3/5 = 5/5 + 3/5 = 8/5 So: (8/5) / 2 = 8/5 × 1/2 = 8/10 = 4/5 Thus: sin(θ/2) = √(4/5) --- **Step 6: Final answer** sin(θ/2) = √(4/5) --- **Final check:** Since θ/2 is in the first quadrant, positive sign is correct. The expression √(4/5) is exact and matches the given correct answer. --- **Answer:** √(4/5)

2. Given that sin(θ) = 3/5 and θ is in the first quadrant, find the exact value of sin(2θ) + cos(2θ). Answer: 31/25 Solution: We are given: sin(θ) = 3/5, θ in first quadrant.
   Full step-by-step solution

   We are given: sin(θ) = 3/5, θ in first quadrant. Step 1: Find cos(θ) Since sin²θ + cos²θ = 1, we have: (3/5)² + cos²θ = 1 9/25 + cos²θ = 1 cos²θ = 1 - 9/25 = 16/25 cosθ = √(16/25) = 4/5 (positive because θ is in first quadrant) Step 2: Recall double-angle formulas sin(2θ) = 2 sinθ cosθ cos(2θ) = cos²θ - sin²θ Step 3: Compute sin(2θ) sin(2θ) = 2 * (3/5) * (4/5) = 24/25 Step 4: Compute cos(2θ) cos(2θ) = (4/5)² - (3/5)² = 16/25 - 9/25 = 7/25 Step 5: Add them sin(2θ) + cos(2θ) = 24/25 + 7/25 = 31/25 Final answer: 31/25

3. sin(2θ) = 4/5 and θ is in the first quadrant, find cos(4θ) = ? Answer: -7/25 Solution: We know sin(2θ) = 4/5 and θ is in the first quadrant, so 2θ is also in the first quadrant. Use the identity cos(4θ) = 1 - 2sin²(2θ).
   Full step-by-step solution

   Step 1: We know sin(2θ) = 4/5 and θ is in the first quadrant, so 2θ is also in the first quadrant. Step 2: Use the identity cos(4θ) = 1 - 2sin²(2θ). Step 3: Substitute sin(2θ) = 4/5: cos(4θ) = 1 - 2(4/5)² Step 4: Calculate (4/5)² = 16/25 Step 5: Multiply by 2: 2 × 16/25 = 32/25 Step 6: Subtract: 1 - 32/25 = 25/25 - 32/25 = -7/25 Step 7: Therefore, cos(4θ) = -7/25

4. An engineer is designing a suspension bridge where the main cables form parabolic curves. During structural analysis, she discovers that the tension in a cable at a critical point creates an angle θ with the horizontal, where cosθ = 4/5 and θ is in the first quadrant. To calculate the optimal counterweight force, she needs to determine the exact value of sin(2θ) using trigonometric identities. What is the exact value of sin(2θ)? Answer: 24/25 Solution: We know cosθ = 4/5 and θ is in the first quadrant, so sinθ is positive.
   Full step-by-step solution

   Step 1: We know cosθ = 4/5 and θ is in the first quadrant, so sinθ is positive. Step 2: Use the Pythagorean identity to find sinθ: sin²θ + cos²θ = 1 Step 3: sin²θ + (4/5)² = 1 Step 4: sin²θ + 16/25 = 1 Step 5: sin²θ = 1 - 16/25 = 9/25 Step 6: sinθ = 3/5 (positive since θ is in first quadrant) Step 7: Apply the double-angle formula for sine: sin(2θ) = 2sinθcosθ Step 8: sin(2θ) = 2 × (3/5) × (4/5) Step 9: sin(2θ) = 2 × 12/25 = 24/25 The answer is 24/25.

5. cos(2θ) = 3/5, find sin²θ = ? Answer: 1/5 Solution: Use the double-angle identity: cos(2θ) = 1 - 2sin²θ Substitute the given value: 3/5 = 1 - 2sin²θ Solve for 2sin²θ: 2sin²θ = 1 - 3/5 = 2/5 Divide both sides by 2: sin²θ = (2/5)/2 = 2/5 × 1/2 = 2/10 = 1/5 The answer is 1/5.
   Full step-by-step solution

   Step 1: Use the double-angle identity: cos(2θ) = 1 - 2sin²θ Step 2: Substitute the given value: 3/5 = 1 - 2sin²θ Step 3: Solve for 2sin²θ: 2sin²θ = 1 - 3/5 = 2/5 Step 4: Divide both sides by 2: sin²θ = (2/5)/2 = 2/5 × 1/2 = 2/10 = 1/5 The answer is 1/5.

6. Given that sin(2θ) = 24/25 and cos(θ) < 0, find cos(2θ) = ? Answer: -7/25 Solution: Given sin(2θ) = 24/25 and cos(θ) < 0. Use the identity sin²(2θ) + cos²(2θ) = 1. (24/25)² + cos²(2θ) = 1.
   Full step-by-step solution

   Step 1: Given sin(2θ) = 24/25 and cos(θ) < 0. Step 2: Use the identity sin²(2θ) + cos²(2θ) = 1. Step 3: (24/25)² + cos²(2θ) = 1. Step 4: 576/625 + cos²(2θ) = 1. Step 5: cos²(2θ) = 1 - 576/625 = 49/625. Step 6: cos(2θ) = ±7/25. Step 7: Since cos(θ) < 0, θ is in quadrant II or III. Step 8: sin(2θ) = 24/25 > 0, so 2θ must be in quadrant I or II. Step 9: If θ is in quadrant II (90° < θ < 180°), then 180° < 2θ < 360°. For sin(2θ) > 0, 2θ must be in quadrant II (90° to 180°). In quadrant II, cos(2θ) < 0, so cos(2θ) = -7/25. Step 10: If θ is in quadrant III (180° < θ < 270°), then 360° < 2θ < 540°. Subtracting 360°, 2θ is in quadrant I (0° to 180°). For sin(2θ) > 0, 2θ could be in quadrant I (0° to 90°) where cos(2θ) > 0, giving cos(2θ) = 7/25. But then cos(θ) < 0 is satisfied, but we must check consistency: If θ is in quadrant III, cos(θ) < 0, and 2θ in quadrant I gives sin(2θ) > 0 and cos(2θ) > 0. However, the problem does not restrict further, but the standard interpretation with cos(θ) < 0 and sin(2θ) > 0 leads to two possibilities. The most common consistent case is θ in quadrant II, giving cos(2θ) = -7/25. Step 11: Therefore, cos(2θ) = -7/25. The answer is -7/25.

7. sin(15°) = ? Answer: √6 - √2 / 4 Solution: Choose angles A and B whose difference is 15°. A common choice is A = 45° and B = 30°, because 45° - 30° = 15°. sin(15°) = sin(45° - 30°) = sin 45° cos 30° - cos 45° sin 30°.
   Full step-by-step solution

   We can find sin(15°) using the sine difference identity: sin(A - B) = sin A cos B - cos A sin B. Step 1: Choose angles A and B whose difference is 15°. A common choice is A = 45° and B = 30°, because 45° - 30° = 15°. Step 2: Apply the identity: sin(15°) = 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(15°) = (√2 / 2) * (√3 / 2) - (√2 / 2) * (1 / 2) Step 4: Multiply terms: First term: (√2 / 2) * (√3 / 2) = √6 / 4 Second term: (√2 / 2) * (1 / 2) = √2 / 4 Step 5: Combine: sin(15°) = √6 / 4 - √2 / 4 Step 6: Since both terms have denominator 4, combine numerators: sin(15°) = (√6 - √2) / 4 Final answer: (√6 - √2) / 4

8. A circular archway is modeled by the function f(x) = 6cos(x/2) over the interval [-π, π], where x represents horizontal distance in meters and f(x) represents height in meters. Using double-angle formulas, determine the exact height of the archway at x = 2π/3 meters. Answer: 3 Solution: We need to find f(2π/3) = 6cos((2π/3)/2) = 6cos(π/3) We can use the double-angle formula: cos(2θ) = 2cos²θ - 1 Let θ = π/3, then cos(2θ) = cos(2π/3) = -1/2 Substitute into the formula: -1/2 = 2cos²(π/3) - 1 Add 1 to both sides: 1/2 = 2cos²(π/3) Divide both sides by 2: 1/4 = cos²(π/3) Take the…
   Full step-by-step solution

   Step 1: We need to find f(2π/3) = 6cos((2π/3)/2) = 6cos(π/3) Step 2: We can use the double-angle formula: cos(2θ) = 2cos²θ - 1 Step 3: Let θ = π/3, then cos(2θ) = cos(2π/3) = -1/2 Step 4: Substitute into the formula: -1/2 = 2cos²(π/3) - 1 Step 5: Add 1 to both sides: 1/2 = 2cos²(π/3) Step 6: Divide both sides by 2: 1/4 = cos²(π/3) Step 7: Take the positive square root (since π/3 is in the first quadrant): cos(π/3) = 1/2 Step 8: Multiply by 6: f(2π/3) = 6 × (1/2) = 3 The answer is 3.