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Trig equations & identities
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The blades of a wind turbine have a diameter of 16m and rotate clockwise at a constant speed, 1 revolution every 4 seconds. The blades are fixed on a shaft such that the tips of the blades are always at least 7m above the ground. The point Q lies at the tip of one of the blades.
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Let h be the height, in meters, of Q above the ground. After t minutes, h is given by h(t)=acos(bt)+c, where a,b,c∈R and a>0.
- [2]
Show that Q starts at the highest possible point.
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Find the values of a, b and c.
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Find all the solutions to the equation sin2θ=√3cosθ, where 0≤θ≤2π.
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The following diagram shows a ferris wheel, which rotates at a constant rate.
The height h, in meters, of point P above the ground is given by h(t)=20cos(bt)+23, where t is the time in minutes.
Find
- [1]
the distance from the lowest point on the wheel to the ground,
- [1]
the diameter of the wheel.
The wheel makes 4 revolutions in an hour.
- [3]
Find the value of b.
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The distance of a passenger from the ground on the London Eye ferris wheel, in meters, is given by
h(t)=71.5−67.5cos(12t°),
where t is how many minutes the passenger has been on the wheel.
- [2]
Find how far above the ground a passenger is when they board the London Eye.
- [2]
Calculate the period of rotation of the London Eye.
The dome atop St. Paul's Cathedral is only visible when passengers are at least 111m above the ground.
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Determine how long the dome is visible during one rotation of the London Eye.
5
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IB: 5
6
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Solve the equation cos4x=sin2x for x∈[0,π]
7
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- [3]
Show that sin2θtanθ≡1−cos2θ.
- [4]
Hence find the value of cot12π, giving your answer in the form p+√q, where p,q∈Z.
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Solve the equation cot2θ+cosecθ=1 for 0≤θ≤2π.
9
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