题目

Problem

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

Solutions relying entirely on calculator technology are not acceptable.

(a) Show that the equation

2\sin\theta(3\cot^2 2\theta-7)=13\sec\theta

can be written as

3\cosec^2 2\theta-13\cosec 2\theta-10=0

(4)

(b) Hence solve, for $0<\theta<\dfrac{\pi}{2}$ , the equation

2\sin\theta(3\cot^2 2\theta-7)=13\sec\theta

giving your answers to 3 significant figures.

(4)

题目中文翻译

本题中你必须写出解题过程的所有步骤。

不接受完全依赖计算器技术的解法。

(a) 证明方程

2\sin\theta(3\cot^2 2\theta-7)=13\sec\theta

可写成

3\cosec^2 2\theta-13\cosec 2\theta-10=0

的形式。

(b) 由此在 $0<\theta<\dfrac{\pi}{2}$ 内解方程

2\sin\theta(3\cot^2 2\theta-7)=13\sec\theta

答案保留 3 位有效数字。

解答

Start with

2\sin\theta(3\cot^2 2\theta-7)=13\sec\theta

Divide both sides by $2\sin\theta$ :

3\cot^2 2\theta-7 =\frac{13\sec\theta}{2\sin\theta}

Since

\sec\theta=\frac1{\cos\theta}

the right hand side becomes

\frac{13}{2\sin\theta\cos\theta}

Using

\sin2\theta=2\sin\theta\cos\theta

we get

3\cot^2 2\theta-7 =13\cosec2\theta

Now use

\cot^2 2\theta=\cosec^2 2\theta-1

3(\cosec^2 2\theta-1)-7=13\cosec2\theta

Expand:

3\cosec^2 2\theta-3-7=13\cosec2\theta

Thus

3\cosec^2 2\theta-10=13\cosec2\theta

Therefore

\boxed{3\cosec^2 2\theta-13\cosec2\theta-10=0}

as required.

From part (a),

3\cosec^2 2\theta-13\cosec2\theta-10=0

Let

u=\cosec2\theta

Then

3u^2-13u-10=0

Factorise:

(3u+2)(u-5)=0

u=-\frac23

u=5

Since $|\cosec2\theta|\ge1$ , reject

u=-\frac23

Thus

\cosec2\theta=5

and hence

\sin2\theta=\frac15

For

0<\theta<\frac{\pi}{2}

we have

0<2\theta<\pi

2\theta=\sin^{-1}\frac15

2\theta=\pi-\sin^{-1}\frac15

Therefore

\theta=\frac12\sin^{-1}\frac15

\theta=\frac12\left(\pi-\sin^{-1}\frac15\right)

Hence

\theta=0.1006\ldots,\ 1.470\ldots

So, to 3 significant figures,

\boxed{\theta=0.101,\ 1.47}