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IAL 2025 May A Q3

A Level / Edexcel / P1

IAL 2025 May A Paper · Question 3

题目

Problem

The graph of the trigonometric function y=f(x)y=f(x), where xx is measured in degrees, is shown.

Figure 2

(a) Write down an expression for f(x)f(x).

(2)

(b) State the number of solutions of each of the following equations for −720∘≤x≤720∘-720^\circ\leq x\leq720^\circ:

(i) f(x)=2f(x)=2

(ii) f(x)=−3f(x)=-3

(2)

解答

(a)

解法一

思路

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从图像读振幅、周期和起点。图像振幅为 33，周期为 360∘360^\circ，并且在 x=0x=0 时位于最低点 −3-3，所以最自然写成 −3cos⁡x-3\cos x。

答题过程

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The graph has amplitude 33 and period 360∘360^\circ.

At x=0x=0, the graph is at its minimum value −3-3, so an expression is

f(x)=−3cos⁡x.\begin{align*} f(x)=-3\cos x. \end{align*}

解法二

思路

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同一条曲线也可以看成正弦曲线平移。因为 sin⁡(x−90∘)=−cos⁡x\sin(x-90^\circ)=-\cos x，所以 3sin⁡(x−90∘)3\sin(x-90^\circ) 与 −3cos⁡x-3\cos x 完全相同。

答题过程

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Using the identity

sin⁡(x−90∘)=−cos⁡x,\begin{align*} \sin(x-90^\circ)=-\cos x, \end{align*}

we can also write

f(x)=3sin⁡(x−90∘).\begin{align*} f(x)=3\sin(x-90^\circ). \end{align*}

(b)

解法一

思路

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用一整个周期来数交点。区间 −720∘-720^\circ 到 720∘720^\circ 长度是 1440∘1440^\circ，也就是 44 个完整周期。

f(x)=2f(x)=2 不是最高或最低值，所以每个周期有 22 个解，总共 88 个。

f(x)=−3f(x)=-3 是最低值，每个完整周期通常有一个最低点；在这个闭区间内，端点和中间点都要数到，所以共有 55 个。

答题过程

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The interval

−720∘≤x≤720∘\begin{align*} -720^\circ\leq x\leq720^\circ \end{align*}

contains 44 full periods of the graph.

For f(x)=2f(x)=2, a horizontal line at y=2y=2 cuts the curve twice in each period. Therefore the number of solutions is

4×2=8.\begin{align*} 4\times2=8. \end{align*}

For f(x)=−3f(x)=-3, the curve reaches its minimum value at

x=−720∘, −360∘, 0∘, 360∘, 720∘.\begin{align*} x=-720^\circ,\,-360^\circ,\,0^\circ,\,360^\circ,\,720^\circ. \end{align*}

So the number of solutions is

5.\begin{align*} 5. \end{align*}