Double Half Angle

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Double Half Angle · Grade 12 · Worksheet 3

1. An architect is designing a modern art installation that features a triangular metal sculpture. The sculpture has sides of length 8 meters and 10 meters with an included angle θ. The architect needs to calculate the exact area of the sculpture, which is given by the formula A = ½(8)(10)sinθ. Through measurements, she determines that cosθ = 3/5. Using double-angle or half-angle formulas, find the exact area of the triangular sculpture. Answer: 32 Solution: Sides: 8 m and 10 m Area formula: A = (1/2) * 8 * 10 * sinθ = 40 * sinθ Given: cosθ = 3/5 Find sinθ using the Pythagorean identity.
   Full step-by-step solution

   We are given: Sides: 8 m and 10 m Included angle: θ Area formula: A = (1/2) * 8 * 10 * sinθ = 40 * sinθ Given: cosθ = 3/5 Step 1: Find sinθ using the Pythagorean identity. sin²θ + cos²θ = 1 sin²θ + (3/5)² = 1 sin²θ + 9/25 = 1 sin²θ = 1 - 9/25 = 16/25 sinθ = ±4/5 Step 2: Determine the sign of sinθ. Since θ is an angle in a triangle, it must be between 0° and 180°, and area is positive, so sinθ > 0. Thus sinθ = 4/5. Step 3: Compute the area. A = 40 * sinθ = 40 * (4/5) A = (40/5) * 4 = 8 * 4 = 32 So the exact area is 32 square meters. The problem mentions using double-angle or half-angle formulas, but here it was simpler to directly find sinθ from cosθ using the Pythagorean identity. If we were to use double-angle: sin²θ = (1 - cos(2θ))/2, but that would require knowing cos(2θ) first, which is unnecessary here. Final answer: 32

2. Kaia is a conservation biologist tracking the flight path of a rare native bird. She observes that the bird's trajectory forms an angle θ with the ground, where θ is in the first quadrant. Using radar data, she determines that cos(2θ) = 11/61. To model the bird's altitude relative to its horizontal distance, Kaia needs the exact value of sin(θ). Using double-angle or half-angle formulas, what is the exact value of sin(θ)? Answer: 5/sqrt(61) Solution: Use the double-angle formula: cos(2θ) = 1 - 2sin²θ Substitute the given value: 11/61 = 1 - 2sin²θ Subtract 1 from both sides: 11/61 - 1 = -2sin²θ Write 1 as 61/61: (11 - 61)/61 = -50/61 = -2sin²θ Divide both sides by -2: sin²θ = (-50/61)/(-2) = 25/61 Take the positive square root (θ in first…
   Full step-by-step solution

   Step 1: Use the double-angle formula: cos(2θ) = 1 - 2sin²θ Step 2: Substitute the given value: 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 = -50/61 = -2sin²θ Step 5: Divide both sides by -2: sin²θ = (-50/61)/(-2) = 25/61 Step 6: Take the positive square root (θ in first quadrant): sinθ = sqrt(25/61) = 5/sqrt(61) The answer is 5/sqrt(61).

3. An architect is designing a modern art sculpture that consists of a triangular metal frame. The frame forms an isosceles triangle where the vertex angle is θ. Through precise measurements, the architect determines that cos(θ) = -3/5. To calculate the exact length of the base supports, she needs to find sin(θ/2). What is the exact value of sin(θ/2)? Answer: 2√5/5 Solution: We are given: cos(θ) = -3/5, and we want sin(θ/2).
   Full step-by-step solution

   We are given: cos(θ) = -3/5, and we want sin(θ/2). Step 1: Recall the half-angle identity for sine: sin(θ/2) = ±√[(1 - cos θ)/2] Step 2: Substitute cos θ = -3/5 into the identity: sin(θ/2) = ±√[(1 - (-3/5))/2] = ±√[(1 + 3/5)/2] = ±√[(8/5)/2] = ±√[8/5 × 1/2] = ±√[8/10] = ±√[4/5] = ±(2/√5) Step 3: Simplify 2/√5 by rationalizing: 2/√5 = (2√5)/5 Step 4: Determine the correct sign. We know θ is the vertex angle of an isosceles triangle. In a triangle, all angles are between 0° and 180°. Here cos θ = -3/5, which is negative, so θ is in the range (90°, 180°). That means θ/2 is in the range (45°, 90°). In that range, sine is positive. So we take the positive root: sin(θ/2) = 2√5/5 Final answer: 2√5/5

4. sin(2θ) = 4/5 and θ is in quadrant I, find cos(4θ) = ? Answer: -7/25 Solution: We know sin(2θ) = 4/5 and θ is in quadrant I, so 2θ is in quadrant I or II. Since sin(2θ) is positive and we're working with standard angles, 2θ is likely in quadrant I where both sine and cosine are positive.
   Full step-by-step solution

   Step 1: We know sin(2θ) = 4/5 and θ is in quadrant I, so 2θ is in quadrant I or II. Since sin(2θ) is positive and we're working with standard angles, 2θ is likely in quadrant I where both sine and cosine are positive. Step 2: Use the Pythagorean identity to find cos(2θ): sin²(2θ) + cos²(2θ) = 1 Step 3: (4/5)² + cos²(2θ) = 1 Step 4: 16/25 + cos²(2θ) = 1 Step 5: cos²(2θ) = 1 - 16/25 = 9/25 Step 6: cos(2θ) = 3/5 (positive since 2θ is in quadrant I) Step 7: Use the double-angle formula for cosine: cos(4θ) = 2cos²(2θ) - 1 Step 8: cos(4θ) = 2(3/5)² - 1 Step 9: cos(4θ) = 2(9/25) - 1 Step 10: cos(4θ) = 18/25 - 25/25 Step 11: cos(4θ) = -7/25 The answer is -7/25.

5. An engineer is designing a suspension bridge where the main cables form parabolic curves. During stress analysis, she needs to calculate the exact tension at a point where the angle between the cable and horizontal is θ. Measurements show that cos(2θ) = -7/25 and θ is in the second quadrant. Using half-angle formulas, determine the exact value of sin(θ) for the tension calculations. Answer: 4/5 Solution: We know cos(2θ) = -7/25 and θ is in the second quadrant (90° < θ < 180°).
   Full step-by-step solution

   Step 1: We know cos(2θ) = -7/25 and θ is in the second quadrant (90° < θ < 180°). Step 2: Use the half-angle formula for sine: sin(θ) = ±√[(1 - cos(2θ))/2] Step 3: Substitute the known value: sin(θ) = ±√[(1 - (-7/25))/2] = ±√[(1 + 7/25)/2] Step 4: Simplify inside the square root: 1 + 7/25 = 25/25 + 7/25 = 32/25 Step 5: Continue: sin(θ) = ±√[(32/25)/2] = ±√[32/50] = ±√[16/25] = ±4/5 Step 6: Since θ is in the second quadrant (90° < θ < 180°), sin(θ) is positive. Step 7: Therefore, sin(θ) = 4/5 Step 8: The exact value of sin(θ) is 4/5.

6. Tane is an astronomer tracking a newly discovered asteroid. He observes that the angle of the asteroid's orbit relative to the ecliptic plane satisfies sin(2θ) = 7/9, where θ is an acute angle in the first quadrant. To calculate the asteroid's exact trajectory, Tane needs to find the exact value of sin(θ) using half-angle formulas. What is the exact value of sin(θ)? Answer: sqrt((9 - sqrt(32))/18) Solution: Use the double-angle identity: sin(2θ) = 2 sinθ cosθ = 7/9. To find sinθ, first find cos(2θ) using the Pythagorean identity: sin²(2θ) + cos²(2θ) = 1. sin²(2θ) = (7/9)² = 49/81.
   Full step-by-step solution

   Step 1: Use the double-angle identity: sin(2θ) = 2 sinθ cosθ = 7/9. Step 2: To find sinθ, first find cos(2θ) using the Pythagorean identity: sin²(2θ) + cos²(2θ) = 1. Step 3: sin²(2θ) = (7/9)² = 49/81. Step 4: So cos²(2θ) = 1 - 49/81 = 32/81. Step 5: Since 2θ is in the first or second quadrant, and sin(2θ) is positive, 2θ could be in first or second quadrant. But since θ is acute (0 < θ < 90°), 2θ < 180°. For sin(2θ) = 7/9, 2θ is in the first quadrant (since 7/9 < 1), so cos(2θ) is positive: cos(2θ) = sqrt(32/81) = sqrt(32)/9. Step 6: Now use the half-angle formula for sine: sinθ = sqrt((1 - cos(2θ))/2). Step 7: Substitute cos(2θ) = sqrt(32)/9: sinθ = sqrt((1 - sqrt(32)/9)/2). Step 8: Simplify inside: 1 - sqrt(32)/9 = (9 - sqrt(32))/9. Step 9: Divide by 2: ((9 - sqrt(32))/9)/2 = (9 - sqrt(32))/(18). Step 10: Take the square root: sinθ = sqrt((9 - sqrt(32))/18). The answer is sqrt((9 - sqrt(32))/18).