Trigression!

Geometry Level 5

If

2 sin ⁡ ( 2 x ) {2\sin(2x)} , sin ⁡ ( 4 x ) {\sin(4x)} and 2 sin ⁡ ( 6 x ) {2\sin(6x)}

are in Arithmetic Progression, then for what values of x x , the above situation holds true. Select the correct option(s).

A: For all real values of x x .

B: For x x is an integer multiple of π 4 \frac{\pi}{4} .

C: For x x is an integer multiple of π 6 \frac{\pi}{6} .

D: For x x is an integer multiple of π 2 \frac{\pi}{2} .

Only C is correct B and D are correct A,B and C are correct Only D is correct B and C are correct B,C and D are correct Only B is correct C and D are correct

Yash Singhal by Yash Singhal , 22, India

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1 solution

Prakhar Gupta
Mar 12, 2015

From the definition of AP:- 2 sin ⁡ 4 x = 2 sin ⁡ 2 x + 2 sin ⁡ 6 x 2\sin4x = 2\sin2x+2\sin6x sin ⁡ 4 x = sin ⁡ 2 x + sin ⁡ 6 x \sin4x=\sin2x+\sin6x sin ⁡ 4 x = 2 sin ⁡ 4 x cos ⁡ 2 x \sin4x = 2\sin4x\cos2x sin ⁡ 4 x = 0 , o r cos ⁡ 2 x = 1 2 \sin4x= 0, or \cos2x = \dfrac{1}{2} x = n π 4 , x = n π ± π 6 x=\dfrac{n\pi}{4} , x = n\pi \pm \dfrac{\pi}{6} Hence only B and D are correct.

2 Helpful 0 Interesting 0 Brilliant 0 Confused

C is also correct ? The solution should be B,C and D.

Parth Bhardwaj - 6 years, 3 months ago

Why C is incorrect? @Prakhar Gupta

Yash Choudhary - 6 years, 2 months ago

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No, C is incorrect because all the integral multiples of π 6 \dfrac{\pi}{6} are not the solution to the given equation.

Prakhar Gupta - 6 years, 2 months ago

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