题目

Problem

Solve, for $0\le x<360^\circ$ , the equation

2\cos 2x=7\cos x

giving your solutions to one decimal place.

(Solutions based entirely on graphical or numerical methods are not acceptable.)

(5)

题目中文翻译

在 $0\le x<360^\circ$ 内解方程

2\cos 2x=7\cos x

答案精确到小数点后 1 位。

（不接受完全基于图像法或数值法的解答。）

解答

We need to solve

2\cos2x=7\cos x

Use the identity

\cos2x=2\cos^2x-1

Then

2(2\cos^2x-1)=7\cos x

Expand:

4\cos^2x-2=7\cos x

Rearrange:

4\cos^2x-7\cos x-2=0

Let

u=\cos x

Then

4u^2-7u-2=0

Factorise:

(4u+1)(u-2)=0

u=-\frac14\quad \text{or}\quad u=2

Since

-1\leq \cos x\leq 1

we reject

u=2

Thus

\cos x=-\frac14

For

0\leq x<360^\circ

the solutions are in the second and third quadrants.

x=104.477\ldots^\circ

x=255.522\ldots^\circ

Therefore, to one decimal place,

\boxed{x=104.5^\circ,\ 255.5^\circ}