State the amplitude and period of \(y=2\sin\left(\dfrac x3\right)-1\).[2]
Sketch the graph of \(y=2\sin\left(\dfrac x3\right)-1\) for \(0\le x\le9\pi\).[3]
It is known that \(\alpha\) and \(\beta\) are the roots of the equation \(2\sin\dfrac x3-1=k\), where \(0<k<1\) and \(\alpha<\beta<3\pi\). Find \(\beta\) in terms of \(\alpha\).[1]
Given that \(2\sin\left(\dfrac x3\right)=p+1\) where \(p\) is a constant, write down the range of values of \(p\) for which the equation has 4 solutions for \(0\le x\le9\pi\).[1]
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Answer:(a) Amplitude 2; period \(6\pi\). (b) See sketch. (c) \(\beta=3\pi-\alpha\). (d) \(-1\le p<1\).