Double Half Angle Worksheets Grade 12

1. Given that sin(2θ) = 9/41 and cos(θ) < 0, find cos(4θ) = ?
2. Given that sin(2θ) = 15/17 and cos(θ) < 0, find cos(2θ) = ?
3. A circular archway is modeled by the function f(x) = 12cos(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.

…and 5 more problems

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.
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)?
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)?

…and 3 more problems

1. Given that cos(2θ) = 2/7 and 0° < θ < 90°, find sin(θ) = ?
2. An architect is designing a modern art sculpture that consists of a triangular metal frame. The frame forms an isosceles triangle with a base of 8 meters and equal sides of 6 meters each. To calculate the optimal lighting angles, the architect needs to determine the exact value of the cosine of half the vertex angle. Using trigonometric identities, find cos(θ/2) where θ is the vertex angle of this triangular frame.
3. Aroha is a geotechnical engineer analyzing the stability of a rock slope. Using laser scanning data, she determines that the angle θ between the slope face and the horizontal satisfies sin(2θ) = 40/41, with θ in the first quadrant. To calculate the factor of safety against sliding, Aroha needs the exact value of tan(θ/2). Using double-angle or half-angle formulas, what is the exact value of tan(θ/2)?

…and 4 more problems