Calculate the value of $ sin(2 heta) $ if $ cos( heta) = frac{3}{5} $ and $ heta $ is in the first quadrant

To find the value of $ \sin(2\theta) $, we use the double angle identity for sine:

$ \sin(2\theta) = 2 \sin(\theta) \cos(\theta) $

Given that $ \cos(\theta) = \frac{3}{5} $, we need to find $ \sin(\theta) $. Since $ \theta $ is in the first quadrant, $ \sin(\theta) $ is positive:

$ \sin(\theta) = \sqrt{1 – \cos^2(\theta)} = \sqrt{1 – \left(\frac{3}{5}\right)^2} = \sqrt{1 – \frac{9}{25}} = \sqrt{\frac{16}{25}} = \frac{4}{5} $

Now we can find $ \sin(2\theta) $:

$ \sin(2\theta) = 2 \cdot \frac{4}{5} \cdot \frac{3}{5} = 2 \cdot \frac{12}{25} = \frac{24}{25} $

To find $ sin(2 heta) $, use the double angle identity:

$ sin(2 heta) = 2 sin( heta) cos( heta) $

Given $ cos( heta) = frac{3}{5} $, we find $ sin( heta) $:

$ sin( heta) = sqrt{1 – cos^2( heta)} = sqrt{1 – left(frac{3}{5}
ight)^2} = sqrt{frac{16}{25}} = frac{4}{5} $

Thus:

$ sin(2 heta) = 2 cdot frac{4}{5} cdot frac{3}{5} = frac{24}{25} $

To find $ sin(2 heta) $, use the formula:

$ sin(2 heta) = 2 sin( heta) cos( heta) $

Since $ cos( heta) = frac{3}{5} $ and $ sin( heta) = frac{4}{5} $:

$ sin(2 heta) = frac{24}{25} $

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