Should I use Triple Angle...

Geometry Level pending

The sum of all solutions to the equation

$\cos 3x+\cos 7x+\cos 11x=0$

over the interval $[0, \pi]$ can be expressed in the form $\frac{a\pi}{b}$ , where $a$ and $b$ are positive coprime integers. What is the value of $ab$ ?

The answer is 22.

1 solution

Chew-Seong Cheong
Dec 20, 2014

$\begin{aligned} \cos {3x} + \cos {7x} + \cos {11x} & = 0 \\ \cos {7x-4x} + \cos {7x} + \cos {7x+4x} & = 0 \\ \cos {7x}\cos{4x} - \sin{7x}\sin{4x} + \cos {7x} + \cos {7x}\cos{4x} + \sin{7x}\sin{4x} & = 0\\ \cos{7x}(2\cos{4x}-1) & =0 \end{aligned}$

$\Rightarrow \begin {cases} \cos{7x} = 0 & \Rightarrow x = \frac{\pi}{14}, \frac{3\pi}{14},\frac{5\pi}{14}, \frac{\pi}{2}, \frac{9\pi}{14}, \frac{11\pi}{14}, \frac{13\pi}{14} \\ 2\cos{4x}-1=0 & \Rightarrow x = \frac{\pi}{12}, \frac{5\pi}{12}, \frac{7\pi}{12}, \frac{11\pi}{12} \end {cases}$

The sum of all solution is $\dfrac {11\pi}{2} = \dfrac {a\pi}{b} \quad \Rightarrow ab = \boxed {22}$

You can just directly use the sum to product formula:

$\cos (11x) + \cos (3x) = 2 \cos \left ( \frac {11x + 3x}{2} \right) \cos \left ( \frac {11x - 3x}{2} \right )$

Pi Han Goh - 6 years, 2 months ago

I wrote a+b , same

NCERT question

U Z - 6 years, 5 months ago

2cos4x + 1 =0 and not 2cos4x - 1 =0. Also pi/2 is not its solution. Yes pi/2 is a solution of cos7x = 0.

vinod trivedi - 4 years, 10 months ago

Thanks, the rest of the solutions were wrong too. I have changed the solutions.

Chew-Seong Cheong - 4 years, 10 months ago

I disagree with you.

http://www.wolframalpha.com/input/?i=cos%283x%29%2Bcos%287x%29%2Bcos%2811x%29%3D0%2C+0%3C%3Dx%3C%3Dpi

Finn Hulse - 6 years, 5 months ago

Should I use Triple Angle...

The answer is 22.

1 solution

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