The decreasing sine branch · source reference

Complete question

(i) Express $\sin x+\sqrt3\cos x$ as $R\sin(x+\alpha)$ with exact positive R and alpha. [1] (ii) Define f by that expression on $\pi/6\le x\le k$. Find the largest exact k for which f has an inverse, define the inverse including its domain, and state the inputs where $ff^{-1}(x)=f^{-1}f(x)$. [5]

[2016/NYJC/Prelim/I/10a]

Printed working

Part (i)

$$\sin x+\sqrt3\cos x=R\sin(x+\alpha)=R\cos\alpha\sin x+R\sin\alpha\cos x.$$

$\tan\alpha=\sqrt3/1\Rightarrow\alpha=\pi/3$ and $R=\sqrt{1^2+(\sqrt3)^2}=2$.

Thus $\sin x+\sqrt3\cos x=2\sin(x+\pi/3)$.

Part (ii): printed working

$f(x)=2\sin(x+\pi/3)$, $\pi/6\le x\le k$. For an inverse, $f$ must be one-one. The largest $k=7\pi/6$.

The source plot shows the closed endpoints $(\pi/6,2)$, $(7\pi/6,-2)$, the x-intercept $(2\pi/3,0)$ and dotted continuations of the unrestricted sine curve.

Let $y=2\sin(x+\pi/3)$, $\pi/6\le x\le7\pi/6$. The source prints

$$x=\sin^{-1}(y/2)-\pi/3,\qquad f^{-1}:x\mapsto\sin^{-1}(x/2)-\pi/3,\quad-2\le x\le2.$$

The printed inverse domain $D_{f^{-1}}=R_f=[-2,2]$ is correct, but the printed inverse rule is the wrong sine branch.

Corrected inverse branch

On the decreasing half-wave, $x+\pi/3=\pi-\sin^{-1}(y/2)$. Thus

$$f^{-1}(x)=\frac{2\pi}3-\sin^{-1}(x/2),\qquad-2\le x\le2.$$

Composite domain comparison

Both identity composites equal $x$ where defined. The printed working gives

$$x\in D_f\cap D_{f^{-1}}=[\pi/6,7\pi/6]\cap[-2,2]=[\pi/6,2].$$

Alternatively $\{x\in\mathbb R:\pi/6\le x\le2\}$. This composite-domain answer is unchanged after correcting the branch.

Teacher margin notes

$D_{ff^{-1}}=D_{f^{-1}}$ and $D_{f^{-1}f}=D_f$. To compare them, intersect their domains: $x\in D_{ff^{-1}}\cap D_{f^{-1}f}\Rightarrow x\in D_{f^{-1}}\cap D_f\Rightarrow x\in[-2,2]\cap[\pi/6,7\pi/6]=[\pi/6,2]$.

Source diagrams

Review qualification

Source discrepancy: solution PDF p.15 prints arcsin(x/2)−π/3. This maps 2 to π/6 but maps −2 to −5π/6, outside the required domain. Porisma shows the corrected branch and retains the original source.

The corrected or qualified authored derivation is retained in the question article.

Provenance

Question page references: 3. Solution page references: 15. The original compilation header and printed page numbers are preserved in the edition notes. Original PDFs and page images remain only in the private QA archive.