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StepMaths Mock Paper 1 · Q10

Paper 1Logic & proofTrigonometryAlgebra

The equation sin⁡(x)=kx\sin(x)=kx has 5 solutions, where k>0k>0. What is the simplest complete set of conditions that kk satisfies when this is true? (You may use the fact that ddxsin⁡x=cos⁡x\dfrac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos x.)

A2π<1−k2k<5π22\pi<\dfrac{\sqrt{1-k^2}}k<\dfrac{5\pi}2 and tan⁡(arccos⁡(k))=arccos⁡(k)\tan(\arccos(k))=\arccos(k)
B2π<1−k2k<5π22\pi<\dfrac{\sqrt{1-k^2}}k<\dfrac{5\pi}2 and tan⁡(1−k2k)=1−k2k\tan\left(\dfrac{\sqrt{1-k^2}}k\right)=\dfrac{\sqrt{1-k^2}}k
C2π<1−k2k<5π22\pi<\dfrac{\sqrt{1-k^2}}k<\dfrac{5\pi}2
Dtan⁡(arccos⁡(k))=arccos⁡(k)\tan(\arccos(k))=\arccos(k)
Etan⁡(1−k2k)=1−k2k\tan\left(\dfrac{\sqrt{1-k^2}}k\right)=\dfrac{\sqrt{1-k^2}}k

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