题目

Problem

Figure 3 shows part of the curve $C_1$ with equation $y=3\sin x$ , where $x$ is measured in degrees.

Figure 3

The point $P$ and the point $Q$ lie on $C_1$ and are shown in Figure 3.

(a) State

(i) the coordinates of $P$ ,

(ii) the coordinates of $Q$ .

(3)

A different curve $C_2$ has equation $y=3\sin x+k$ , where $k$ is a constant.

The curve $C_2$ has a maximum $y$ value of $10$ .

The point $R$ is the minimum point on $C_2$ with the smallest positive $x$ coordinate.

(b) State the coordinates of $R$ .

(2)

解答

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$y=3\sin x$ 的最大值是 $3$ ，第一次出现在 $x=90^\circ$ 。从图上看， $Q$ 是下一次与 $x$ 轴相交的点，对应 $x=540^\circ$ 。

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The maximum point shown is

\begin{align*} P=(90^\circ,3). \end{align*}

The next intercept shown is

\begin{align*} Q=(540^\circ,0). \end{align*}

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$3\sin x$ 的最大值是 $3$ 。如果 $3\sin x+k$ 的最大值是 $10$ ，则整体上移了 $7$ 。最小值是 $-3+7=4$ ，最小点第一次出现在 $x=270^\circ$ 。

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Since the maximum value is $10$ ,

\begin{align*} 3+k=10 \quad\Rightarrow\quad k=7. \end{align*}

The minimum value is

\begin{align*} -3+7=4. \end{align*}

The smallest positive $x$ coordinate of a minimum point is $270^\circ$ , so

\begin{align*} R=(270^\circ,4). \end{align*}