”Solve
Samuel Scott
Given:
$ \sin(\theta)\cos(\theta) = \frac{1}{4} $
Using the double-angle identity:
$ \sin(2\theta) = 2\sin(\theta)\cos(\theta) $
We have:
$ \sin(2\theta) = 2 \times \frac{1}{4} = \frac{1}{2} $
Thus:
$ 2\theta = \sin^{-1}(\frac{1}{2}) $
Giving:
$ 2\theta = \frac{\pi}{6} \text{ or } \frac{5\pi}{6} $
Hence:
$ \theta = \frac{\pi}{12} \text{ or } \frac{5\pi}{12} $
Checking the interval $ 0 \leq \theta < 2\pi $:
The possible solutions are:
$ \theta = \frac{\pi}{12}, \frac{5\pi}{12} \text{ or } \frac{13\pi}{12}, \frac{17\pi}{12} $
Answer 2
Michael Moore
Given:
$ sin( heta)cos( heta) = frac{1}{4} $
Using the double-angle identity:
$ sin(2 heta) = 2sin( heta)cos( heta) $
We have:
$ sin(2 heta) = frac{1}{2} $
Thus:
$ 2 heta = frac{pi}{6} ext{ or } frac{5pi}{6} $
Hence:
$ heta = frac{pi}{12} ext{ or } frac{5pi}{12} $
Answer 3
Isabella Walker
Given:
$ sin( heta)cos( heta) = frac{1}{4} $
Using the double-angle identity:
$ sin(2 heta) = 2sin( heta)cos( heta) $
We have:
$ sin(2 heta) = frac{1}{2} $
Thus:
$ 2 heta = frac{pi}{6} ext{ or } frac{5pi}{6} $
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