Expression rewritten as a quadratic in cosecant and the resulting equation solved using arcsine
Expression rewritten as a quadratic in cosecant and the resulting equation solved using arcsine
IB Year 6 | Grade 12
6 marks
The function \(\mathrm{f}\) is given by \(\mathrm{f}(x)=3\cot^2x+7\,\mathrm{cosec}\,x+4\cos2x+8\sin^2x-21\).
- Show that \(\mathrm{f}(x)\) can be written in terms of \(\mathrm{cosec}\,x\) only, and give this expression.[2]
- Solve the equation \(\mathrm{f}(x)=0\) for \(0<x<\pi\), giving your answers exactly in terms of \(\arcsin\).[4]
Solution:
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Answer:(a) \(3\,\mathrm{cosec}^2x+7\,\mathrm{cosec}\,x-20\) (b) \(x=\arcsin\dfrac35\) or \(x=\pi-\arcsin\dfrac35\)
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