Question

What are the values of sin 4x = -2 sin 2x?

Answer 1

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Solve the trigonometric equation:

sin 4x = – 2 · sin 2x

sin 4x + 2 · sin 2x = 0          (i)


Use the double-angle formula for sin 4x:

sin 4x = 2 · sin 2x · cos 2x


Substitute that into (i), and you have

2 · sin 2x · cos 2x + 2 · sin 2x = 0          (ii)


Now, take out the common factor 2 · sin 2x:

2 · sin 2x · (cos 2x + 1) = 0

sin 2x · (cos 2x + 1) = 0


If the product above equals zero, then either sin 2x or (cos 2x + 1) must be zero. So,

sin 2x = 0     or     cos 2x + 1 = 0

sin 2x = 0     or     cos 2x = –1


Solving them separately:

•  sin 2x = 0

2x = k · π
 
         k · π 
x  =  ———
            2

where k is an integer    (all integer multiples of π/2).


•  cos 2x = – 1

2x = π + k · 2π

2x = (1 + 2k) · π
  
         (1 + 2k) · π
x  =  ——————
                 2

where k is an integer     (all odd integer multiples of π/2).


But notice that the solution for sin 2x = 0 also includes those particular cases that we found when solving cos 2x = – 1.


Therefore, the solution set is simply

$\mathsf{S=\left\{x\in\mathbb{R}:~~x=\dfrac{k\cdot \pi}{2},~~k\in\mathbb{Z}\right\}}$

(all integer multiples of π/2).


I hope this helps. =)


Tags:  solve trigonometric equation double angle formula factor sine cosine sin cos trig trigonometry