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A coffee filter is made from a sector of radius 2cm and central angle θ. The filter can be folded into a cone of slant height R.
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Show that the volume of the filter is V=3π2θ2√4π2−θ2.
Hence show that dθdV=3π2√4π2−θ2θ(8π2−3θ2).
Find the exact value of θ that maximizes V.
The value of θ is increasing at a rate of 0.2 radians per second.
Find the rate of change in the volume when θ=π.
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Let w=cosθ+isinθ, for 43π<θ<45π.
Using the binomial theorem, write down the expansion w3.
Using your result from (a), show that
(i) cos3θ=4cos3θ−3cosθ
(ii) sin3θ=3sinθ−4sin3θ
Hence, or otherwise, show that 1−3tan2A3tanA−tan3A≡tan3A.
Using the identity demonstrated in (c), find the exact value of tan3θ, given that sinθ=53.
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