Question

What is the answer to "sin2x=sinx" ?

Answer 1

`\bold{x=\frac\pi3+2k\pi\quad \vee\quad x=\frac{5\pi}3+2k\pi\quad \vee\quad x=k\pi\ ,\quad k\in\mathbb Z}`

Step-by-step explanation:

`\sin 2x=\sin x\\\\\sin2x - \sin x=0\\\\2\sin x\cos x-\sin x=0\\\\(2\cos x-1)\sin x=0\\\\2\cos x-1=0\qquad\qquad\qquad\qquad or\qquad \sin x=0\\\\2\cos x=1\quad\qquad\qquad\qquad\qquad\ or\qquad \sin x=0\\\\\cos x=\frac12\qquad\qquad\qquad\qquad\qquad or\qquad \sin x=0\\\\x=\frac\pi3+2k\pi\quad \vee\ \ x=\frac{5\pi}3+2k\pi\quad\ \ \vee\ \ x=0+k\pi\ ,\quad k\in\mathbb Z`

Answer 2

Think of the double angle formula for

sin

2

x

Step-by-step explanation:

sin

2

x

=

sin

x

2

sin

x

cos

x

=

sin

x

2

sin

x

cos

x

−

sin

x

=

0

sin

x

(

2

cos

x

−

1

)

=

0

Solution A:

sin

x

=

0

⇒

x

=

k

π

,

k

∈

Z

Solution B:

2

cos

x

=

1

⇒

cos

x

=

1

2

,

x

=

±

π

3

+

2

k

π

=

π

3

(

6

k

±

1

)

,

k

∈

Z

∴

x

=

k

π

or

x

=

π

3

(

6

k

±

1

)

,

k

∈

Z

Step-by-step explanation: