Double Half Angle

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Answer Key & Explanations

Double Half Angle · Grade 12 · Worksheet 2

1. A large symmetrical geometric sculpture consists of two identical parabolic arches that intersect at their peaks. The angle between the two arches at the peak is 150 degrees. Using a half-angle formula, determine the exact value of sin(75 degrees), which represents the sine of half the angle between one arch and the vertical axis. Answer: (sqrt(6) + sqrt(2)) / 4 Solution: Recognize that 75 degrees is half of 150 degrees. Use the half-angle formula for sine: sin(theta/2) = sqrt((1 - cos(theta))/2). Let theta = 150 degrees.
   Full step-by-step solution

   Step 1: Recognize that 75 degrees is half of 150 degrees. Use the half-angle formula for sine: sin(theta/2) = sqrt((1 - cos(theta))/2). Step 2: Let theta = 150 degrees. Then sin(75 degrees) = sqrt((1 - cos(150 degrees))/2). Step 3: Find cos(150 degrees). Since 150 degrees is in the second quadrant, cos(150 degrees) = -cos(30 degrees) = -sqrt(3)/2. Step 4: Substitute: sin(75 degrees) = sqrt((1 - (-sqrt(3)/2))/2) = sqrt((1 + sqrt(3)/2)/2). Step 5: Simplify inside the square root: (1 + sqrt(3)/2)/2 = (2/2 + sqrt(3)/2)/2 = ((2 + sqrt(3))/2)/2 = (2 + sqrt(3))/4. Step 6: So sin(75 degrees) = sqrt((2 + sqrt(3))/4) = sqrt(2 + sqrt(3))/2. Step 7: Express sqrt(2 + sqrt(3)) in a simpler form. Note that (sqrt(6) + sqrt(2))^2 / 4 = (6 + 2 + 2*sqrt(12))/4 = (8 + 4*sqrt(3))/4 = 2 + sqrt(3). Thus sqrt(2 + sqrt(3)) = (sqrt(6) + sqrt(2))/2. Step 8: Therefore sin(75 degrees) = ((sqrt(6) + sqrt(2))/2) / 2 = (sqrt(6) + sqrt(2))/4. Since 75 degrees is in the first quadrant, sine is positive. The answer is (sqrt(6) + sqrt(2))/4.

2. A civil engineer is designing a suspension bridge where the main cables form parabolic curves. During structural analysis, she determines that the angle between the cable and the horizontal support at a critical point satisfies tan(θ) = 4/3, with θ in the first quadrant. To calculate the stress distribution in the cable material, she needs to find the exact value of sin(2θ) using trigonometric identities. What is the exact value of sin(2θ)? Answer: 24/25 Solution: We know tan(θ) = 4/3, which means opposite/adjacent = 4/3 Using the Pythagorean theorem, hypotenuse = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5 Therefore, sin(θ) = opposite/hypotenuse = 4/5 cos(θ) = adjacent/hypotenuse = 3/5 Using the double-angle formula: sin(2θ) = 2sin(θ)cos(θ) Substitute…
   Full step-by-step solution

   Step 1: We know tan(θ) = 4/3, which means opposite/adjacent = 4/3 Step 2: Using the Pythagorean theorem, hypotenuse = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5 Step 3: Therefore, sin(θ) = opposite/hypotenuse = 4/5 Step 4: cos(θ) = adjacent/hypotenuse = 3/5 Step 5: Using the double-angle formula: sin(2θ) = 2sin(θ)cos(θ) Step 6: Substitute the values: sin(2θ) = 2 × (4/5) × (3/5) Step 7: Calculate: sin(2θ) = 2 × 12/25 = 24/25 The answer is 24/25.

3. Emma is an architect designing a parabolic arch for a new botanical garden entrance. The arch's shape is defined by a quadratic function, and at a specific point on the arch, the angle of inclination θ (measured from the horizontal) satisfies cos(2θ) = -1/8, with 2θ in the second quadrant. To calculate the precise curvature of the arch at this point, Emma needs to find the exact value of cos(θ). Using double-angle or half-angle formulas, what is the exact value of cos(θ)? Answer: sqrt(7/16) or sqrt(7)/4 Solution: We are given cos(2θ) = -1/8, and 2θ is in the second quadrant. We need cos(θ). Use the double-angle formula for cosine: cos(2θ) = 2cos²(θ) - 1.
   Full step-by-step solution

   Step 1: We are given cos(2θ) = -1/8, and 2θ is in the second quadrant. We need cos(θ). Step 2: Use the double-angle formula for cosine: cos(2θ) = 2cos²(θ) - 1. Step 3: Substitute the given value: -1/8 = 2cos²(θ) - 1. Step 4: Add 1 to both sides: -1/8 + 1 = 2cos²(θ). Step 5: -1/8 + 8/8 = 7/8, so 7/8 = 2cos²(θ). Step 6: Divide both sides by 2: cos²(θ) = 7/16. Step 7: Take the square root: cos(θ) = ± sqrt(7/16) = ± sqrt(7)/4. Step 8: Determine the sign: Since 2θ is in the second quadrant, 2θ is between 90° and 180°, so θ is between 45° and 90°, which is the first quadrant. In the first quadrant, cosine is positive. Step 9: Therefore, cos(θ) = sqrt(7)/4. The answer is sqrt(7)/4.

4. 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.

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. Olivia is a civil engineer designing a curved pedestrian bridge. The bridge's main support cable forms a parabolic curve, and at a specific point, the angle of inclination θ (measured from the horizontal) satisfies cos(2θ) = -1/3, with 2θ in the second quadrant. To determine the exact force distribution along the cable, Olivia needs to find the exact value of sin(θ). Using double-angle or half-angle formulas, what is the exact value of sin(θ)? Answer: sqrt(6)/3 Solution: Use the double-angle formula: cos(2θ) = 1 - 2sin²θ Substitute the given value: -1/3 = 1 - 2sin²θ Subtract 1 from both sides: -1/3 - 1 = -2sin²θ Simplify the left side: -1/3 - 3/3 = -4/3 = -2sin²θ Divide both sides by -2: sin²θ = 2/3 Take the square root: sinθ = sqrt(2/3) = sqrt(2)/sqrt(3)…
   Full step-by-step solution

   Step 1: Use the double-angle formula: cos(2θ) = 1 - 2sin²θ Step 2: Substitute the given value: -1/3 = 1 - 2sin²θ Step 3: Subtract 1 from both sides: -1/3 - 1 = -2sin²θ Step 4: Simplify the left side: -1/3 - 3/3 = -4/3 = -2sin²θ Step 5: Divide both sides by -2: sin²θ = 2/3 Step 6: Take the square root: sinθ = sqrt(2/3) = sqrt(2)/sqrt(3) Step 7: Rationalize: sqrt(2)/sqrt(3) = sqrt(6)/3 Step 8: Since 2θ is in the second quadrant (between 90° and 180°), θ is between 45° and 90°, so θ is in the first quadrant and sinθ is positive. The answer is sqrt(6)/3.