The function $\mathrm{f}$ is defined by $$\mathrm{f}(x)=a\sin2x+b\cos2x+c,\qquad0\le x\le\pi,$$ wh

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f}(x)=a\sin2x+b\cos2x+c,\qquad0\le x\le\pi,$$ where $a,b,c$ are constants. The curve $y=\mathrm{f}(x)$ passes through $(0,3)$ and has a stationary point at $\left(\frac\pi6,4\right)$. Form three linear equations in $a,b,c$, and hence find their exact values.[5] Express $\mathrm{f}(x)$ in the form $R\sin(2x+\alpha)+c$, where $R>0$ and $0<\alpha<\frac\pi2$.[2] Hence solve $\mathrm{f}(x)=3$, giving all answers in exact form.[3]

Accepted Answer

(i) $a=\sqrt3,\ b=1,\ c=2$ (ii) $\mathrm{f}(x)=2\sin\left(2x+\frac\pi6\right)+2$ (iii) $x=0,\frac\pi3,\pi$

A trigonometric curve determined by a stationary point

A trigonometric curve determined by a stationary point

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