Given that $ csc( heta) = 2 $ and $ heta $ lies in the second quadrant, find the exact value of $ heta $ and verify using trigonometric identities.

Given:

$ \csc(\theta) = 2 $

Since \( \csc(\theta) = \frac{1}{\sin(\theta)} \), we get:

$ \sin(\theta) = \frac{1}{2} $

In the second quadrant, angle \( \theta \) where \( \sin(\theta) = \frac{1}{2} \) is:

$ \theta = 180^\circ – 30^\circ = 150^\circ $

Converting to radians:

$ \theta = \pi – \frac{\pi}{6} = \frac{5\pi}{6} $

Verification:

$ \csc(\frac{5\pi}{6}) = \frac{1}{\sin(\frac{5\pi}{6})} = \frac{1}{\frac{1}{2}} = 2 $

Thus, the exact value of \( \theta \) is:

$ \boxed{\frac{5\pi}{6}} $

Given:

$ csc( heta) = 2 $

We know:

$ csc( heta) = frac{1}{sin( heta)} $

Thus:

$ sin( heta) = frac{1}{2} $

In the second quadrant, ( heta ) is:

$ heta = 180^circ – 30^circ = 150^circ $

Converting to radians:

$ heta = frac{5pi}{6} $

To verify:

$ csc(frac{5pi}{6}) = csc(150^circ) = 2 $

Thus, the exact value of ( heta ) is:

$ oxed{frac{5pi}{6}} $

Given:

$ csc( heta) = 2 $

Since:

$ csc( heta) = frac{1}{sin( heta)} $

Then:

$ sin( heta) = frac{1}{2} $

In the second quadrant:

$ heta = 150^circ = frac{5pi}{6} $

Verify:

$ csc(150^circ) = 2 $

The exact value of ( heta ) is:

$ oxed{frac{5pi}{6}} $

Start Using PopAi Today