题目

Problem

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

f(x)=7\cos x-24\sin x

(a) Express $f(x)$ in the form $R\cos(x+\alpha)$ where $R$ and $\alpha$ are constants, $R>0$ and $0<\alpha<\dfrac{\pi}{2}$ .

Give the exact value of $R$ and give the value of $\alpha$ , in radians, to 3 decimal places.

(3)

g(x)=\frac{5}{90-3f(2x)}

(b) Using the answer to part (a), find

(i) the minimum value of $g(x)$ , giving your answer as a fully simplified fraction,

(ii) the smallest positive value of $x$ for which this minimum value occurs, giving your answer to 3 decimal places.

(4)

题目中文翻译

本题必须写出全部解题步骤。

不接受完全依赖计算器技术的解法。

f(x)=7\cos x-24\sin x

(a) 将 $f(x)$ 写成 $R\cos(x+\alpha)$ 的形式，其中 $R,\alpha$ 为常数，且 $R>0,\ 0<\alpha<\dfrac{\pi}{2}$ 。

写出 $R$ 的精确值，并将 $\alpha$ 的弧度值保留到小数点后 3 位。

g(x)=\frac{5}{90-3f(2x)}

(b) 利用(a)中的结果，求

(i) $g(x)$ 的最小值，并将答案写成最简分数；

(ii) 取得该最小值时最小的正 $x$ 值，答案保留到小数点后 3 位。

解答

We want

7\cos x-24\sin x=R\cos(x+\alpha)

Expand the right side:

R\cos(x+\alpha)=R\cos x\cos\alpha-R\sin x\sin\alpha

Compare coefficients:

R\cos\alpha=7

and

R\sin\alpha=24

Square and add:

R^2=7^2+24^2=49+576=625

R=25

Also,

\tan\alpha=\frac{24}{7}

Hence

\alpha=\arctan\left(\frac{24}{7}\right)=1.287\ldots

Therefore

\boxed{f(x)=25\cos(x+1.287\ldots)}

with

\boxed{R=25,\quad \alpha=1.287}

Using part (a),

f(2x)=25\cos(2x+\alpha)

g(x)=\frac{5}{90-75\cos(2x+\alpha)}

For $g(x)$ to be as small as possible, its denominator must be as large as possible.

The maximum denominator occurs when

\cos(2x+\alpha)=-1

Then

90-75(-1)=165

So the minimum value is

\frac{5}{165}=\frac{1}{33}

Therefore

\boxed{\text{minimum value of }g(x)=\frac{1}{33}}

For the smallest positive $x$ giving this minimum,

2x+\alpha=\pi

x=\frac{\pi-\alpha}{2}

Using

\alpha=1.2870\ldots

we get

x=0.927\ldots

Therefore

\boxed{x=0.927}

to 3 decimal places.