Question

find theta if 2 sin 2 theta =√ 3

Answer 1

Given :

2 sin 2θ = √3

==> sin 2θ = √3 / 2

==> sin 2θ = sin π/3

Note that :

In the first quadrant all ratios are positive

In the second quadrant sin,cosec are positive

So:

In the first quadrant : sin 2θ = π/3

==> θ = π/6

In the second quadrant : sin 2θ = sin (π-π/3)

==> sin( 2π/3)

2θ = 2π/3

==> θ = π/3

Let n be the number of solutions.

So :

θ = π/6 ± πn and

θ = π/3 ± πn

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Answer 2

Answer:

$\theta=\frac{1}{2}n\pi+(-1)^n\frac{\pi}{6}$

Step-by-step explanation:

Given,

$2\sin2\theta=\sqrt{3}\\\;\\\sin2\theta=\frac{\sqrt{3}}{2}\\\;\\\sin2\theta=\sin\frac{\pi}{3}\\\;\\2\theta=n\pi+(-1)^n\frac{\pi}{3}\\\;\\\theta=\frac{1}{2}n\pi+(-1)^n\frac{\pi}{6}$

$Note:\;if,\\\;\\\sin\theta=\sin\alpha\\\;\\\theta=n\pi+(-1)^n\alpha$