Double Half Angle

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Double Half Angle · Grade 12 · Worksheet 2

1. A circular archway is modeled by the function f(x) = 12cos(x/3) over the interval [-3π/2, 3π/2], 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 = π/2 meters. Answer: 6√3 Solution: The function is f(x) = 12cos(x/3). We need to find f(π/2) = 12cos((π/2)/3) = 12cos(π/6). Let 2θ = π/3, then θ = π/6.
   Full step-by-step solution

   Step 1: The function is f(x) = 12cos(x/3). We need to find f(π/2) = 12cos((π/2)/3) = 12cos(π/6). Step 2: We can use the double-angle identity cos(2θ) = 2cos²θ - 1 to find cos(π/6). Step 3: Let 2θ = π/3, then θ = π/6. The identity becomes cos(π/3) = 2cos²(π/6) - 1. Step 4: We know cos(π/3) = 1/2, so 1/2 = 2cos²(π/6) - 1. Step 5: Add 1 to both sides: 1/2 + 1 = 2cos²(π/6) → 3/2 = 2cos²(π/6). Step 6: Divide both sides by 2: cos²(π/6) = 3/4. Step 7: Take the positive square root (since π/6 is in the first quadrant): cos(π/6) = √(3/4) = √3/2. Step 8: Now calculate f(π/2) = 12 × (√3/2) = 6√3. The answer is 6√3.

2. Given that sin(2θ) = 24/25 and cos(θ) > 0, find cos(4θ) = ? Answer: −527/625 Solution: Use the double-angle identity for cosine: cos(4θ) = 1 − 2sin²(2θ). Substitute sin(2θ) = 24/25: cos(4θ) = 1 − 2(24/25)². Compute (24/25)² = 576/625.
   Full step-by-step solution

   Step 1: Use the double-angle identity for cosine: cos(4θ) = 1 − 2sin²(2θ). Step 2: Substitute sin(2θ) = 24/25: cos(4θ) = 1 − 2(24/25)². Step 3: Compute (24/25)² = 576/625. Step 4: Multiply by 2: 2 × 576/625 = 1152/625. Step 5: Subtract from 1: 1 − 1152/625 = 625/625 − 1152/625 = −527/625. Step 6: The condition cos(θ) > 0 does not affect the sign of cos(4θ) because the identity gives the exact value directly. The answer is −527/625.

3. sin(2θ) = 3/5, cos(2θ) = 4/5, find sin²(θ) = ? Answer: 1/10 Solution: sin(2θ) = 3/5 cos(2θ) = 4/5 We want sin²(θ).
   Full step-by-step solution

   We are given: sin(2θ) = 3/5 cos(2θ) = 4/5 We want sin²(θ). --- **Step 1: Recall the double-angle identity for cos(2θ)** cos(2θ) = 1 − 2 sin²(θ) --- **Step 2: Substitute the given value of cos(2θ)** 4/5 = 1 − 2 sin²(θ) --- **Step 3: Solve for sin²(θ)** 2 sin²(θ) = 1 − 4/5 2 sin²(θ) = 1/5 sin²(θ) = (1/5) / 2 sin²(θ) = 1/10 --- **Step 4: Conclusion** The answer is 1/10.

4. An architect is designing a modern art sculpture that consists of a triangular metal frame. The frame forms an isosceles triangle where the base angles are both θ. Through precise measurements, it's determined that cos(2θ) = -7/25. The architect needs to calculate the exact value of sin(θ) to determine the optimal placement of lighting elements. What is the exact value of sin(θ)? Answer: 4/5 Solution: We have an isosceles triangle with base angles both equal to θ. Given: cos(2θ) = -7/25. We are to find sin(θ).
   Full step-by-step solution

   Let's solve step by step. --- **Step 1: Understand the given information** We have an isosceles triangle with base angles both equal to θ. Given: cos(2θ) = -7/25. We are to find sin(θ). --- **Step 2: Recall the double-angle identity for cosine** The identity is: cos(2θ) = 1 - 2 sin²(θ) --- **Step 3: Substitute the given value into the identity** cos(2θ) = -7/25 So: 1 - 2 sin²(θ) = -7/25 --- **Step 4: Solve for sin²(θ)** 1 - 2 sin²(θ) = -7/25 Add 2 sin²(θ) to both sides: 1 = 2 sin²(θ) - 7/25 Add 7/25 to both sides: 1 + 7/25 = 2 sin²(θ) 1 = 25/25, so: 25/25 + 7/25 = 32/25 = 2 sin²(θ) Thus: 2 sin²(θ) = 32/25 Divide both sides by 2: sin²(θ) = 32/50 Simplify: 32/50 = 16/25 --- **Step 5: Take the square root** sin(θ) = √(16/25) = 4/5 or -4/5 --- **Step 6: Determine the correct sign** Since θ is a base angle of a triangle, 0° < θ < 90°, so sin(θ) > 0. Therefore: sin(θ) = 4/5 --- **Final answer:** 4/5

5. cos(2θ) = 1/3, 0° < θ < 90°, sin(4θ) = ? Answer: 4√2/9 Solution: We are given cos(2θ) = 1/3 and θ is in the first quadrant.
   Full step-by-step solution

   Step 1: We are given cos(2θ) = 1/3 and θ is in the first quadrant. Step 2: Use the Pythagorean identity to find sin(2θ): sin²(2θ) + cos²(2θ) = 1 Step 3: sin²(2θ) = 1 - (1/3)² = 1 - 1/9 = 8/9 Step 4: Since θ is in the first quadrant, 2θ is also in the first quadrant, so sin(2θ) is positive: sin(2θ) = √(8/9) = 2√2/3 Step 5: Use the double-angle identity for sin(4θ): sin(4θ) = 2sin(2θ)cos(2θ) Step 6: Substitute the known values: sin(4θ) = 2 × (2√2/3) × (1/3) = 4√2/9 Step 7: The final answer is 4√2/9.

6. Matiu is a structural engineer analyzing the forces on a triangular roof truss. The truss forms an isosceles triangle where the vertex angle at the peak is θ. Through precise laser measurements, he determines that cos(2θ) = -11/61, and that 2θ is in the second quadrant. To calculate the optimal distribution of load-bearing cables, Matiu needs the exact value of sin(θ). Using double-angle or half-angle formulas, what is the exact value of sin(θ)? Answer: 6/sqrt(61) Solution: Use the double-angle formula cos(2θ) = 1 - 2sin²θ. Substitute cos(2θ) = -11/61: -11/61 = 1 - 2sin²θ. Subtract 1 from both sides: -11/61 - 1 = -2sin²θ.
   Full step-by-step solution

   Step 1: Use the double-angle formula cos(2θ) = 1 - 2sin²θ. Step 2: Substitute cos(2θ) = -11/61: -11/61 = 1 - 2sin²θ. Step 3: Subtract 1 from both sides: -11/61 - 1 = -2sin²θ. Step 4: Write 1 as 61/61: -11/61 - 61/61 = -72/61 = -2sin²θ. Step 5: Multiply both sides by -1: 72/61 = 2sin²θ. Step 6: Divide both sides by 2: 36/61 = sin²θ. Step 7: Take the square root: sinθ = ± sqrt(36/61) = ± 6/sqrt(61). Step 8: Since 2θ is in the second quadrant, θ is in the first quadrant (because if 90° < 2θ < 180°, then 45° < θ < 90°). Thus sinθ is positive. Step 9: Therefore, sinθ = 6/sqrt(61). The answer is 6/sqrt(61).

7. cos(2θ) = 2/3, 0° < θ < 90°, find sin(θ) = ? Answer: √6/6 Solution: Use the identity cos(2θ) = 1 - 2sin²(θ). Substitute the given value: 2/3 = 1 - 2sin²(θ). Rearrange: 2sin²(θ) = 1 - 2/3 = 1/3.
   Full step-by-step solution

   Step 1: Use the identity cos(2θ) = 1 - 2sin²(θ). Step 2: Substitute the given value: 2/3 = 1 - 2sin²(θ). Step 3: Rearrange: 2sin²(θ) = 1 - 2/3 = 1/3. Step 4: Divide by 2: sin²(θ) = 1/6. Step 5: Take the square root: sin(θ) = ±√(1/6) = ±1/√6 = ±√6/6. Step 6: Since 0° < θ < 90°, θ is in quadrant I, where sine is positive. Step 7: Therefore, sin(θ) = √6/6. The answer is √6/6.

8. cos(2θ) = 3/5, 0° < θ < 90°, sin(θ) = ? Answer: √(1/5) Solution: We are given: cos(2θ) = 3/5, with 0° < θ < 90°, and we want sin(θ).
   Full step-by-step solution

   We are given: cos(2θ) = 3/5, with 0° < θ < 90°, and we want sin(θ). Step 1: Recall the double-angle identity for cosine in terms of sine: cos(2θ) = 1 - 2 sin²(θ) Step 2: Substitute the given value into the identity: 1 - 2 sin²(θ) = 3/5 Step 3: Solve for sin²(θ): 1 - 3/5 = 2 sin²(θ) (5/5 - 3/5) = 2 sin²(θ) 2/5 = 2 sin²(θ) Step 4: Divide both sides by 2: sin²(θ) = (2/5) / 2 = (2/5) * (1/2) = 2/10 = 1/5 Step 5: Take the square root: sin(θ) = ±√(1/5) Step 6: Determine the correct sign using the given range for θ: Since 0° < θ < 90°, θ is in the first quadrant, where sine is positive. Therefore: sin(θ) = √(1/5) Final answer: √(1/5)