Multiple Angle Trigonometry Worksheets Grade 12

1. A Ferris wheel with a diameter of 40 meters completes one full revolution every 2 minutes. The height of a passenger above the ground can be modeled by the function h(t) = 20 + 20sin(πt - π/2), where t is time in minutes after boarding. At what times during the first 4 minutes will the passenger be exactly 30 meters above the ground?
2. Charlotte is analyzing the path of a laser beam reflecting off a series of mirrors. The beam's angle relative to a reference line, measured in degrees, is given by the equation cos(3θ + 15°) = 0.5, where θ is the time in seconds. Find all times θ in the interval [0°, 360°) that satisfy this equation.
3. 2cos²(3x) - 1 = 0 for x ∈ [0, π]

…and 4 more problems

1. An engineer is designing a roller coaster track where the vertical position of the track follows the function y(x) = 3sin(2x) + 4cos(2x), where x is the horizontal distance in meters from the starting point. To ensure proper banking at turns, she needs to find all points where the track reaches exactly 2 meters above the reference line. Solve the equation 3sin(2x) + 4cos(2x) = 2 for x in the interval [0, 2π].
2. A triangular garden is bounded by three straight paths. The paths form a triangle with vertices at coordinates A(0,0), B(8,0), and C(4,6). A circular fountain is to be placed at the incenter of this triangle. What are the coordinates of the fountain's center?
3. An engineer is designing a suspension bridge where the cable shape follows the curve y = 3sin(2x) + 4cos(2x). To determine the maximum tension points, she needs to find all values of x between 0 and 2π where the cable reaches its highest point. Solve the trigonometric equation to find these critical points.

…and 5 more problems

1. An oceanographer is studying tidal patterns in a coastal bay. The water depth D(t) in meters is modeled by the function D(t) = 3.5 + 2.8cos(πt/6) + 1.2sin(πt/6), where t is time in hours after midnight. To determine when boats with a 4.2 meter draft can safely enter the harbor, she needs to find all times between 6:00 AM and 6:00 PM when the water depth is exactly 5.1 meters. Solve the trigonometric equation to find these times.
2. 2cos²(2x) - 1 = 0 for x ∈ [0, 2π]
3. Emma is observing a laser beam reflecting off a rotating mirror. The angle of the reflected beam, measured in degrees from a fixed reference line, is given by θ(t) = 15 sin(5t + 10°) + 25, where t is time in seconds. At what times t in the interval [0, 360) seconds does the reflected beam make an angle of exactly 10° with the reference line?

…and 4 more problems