Find the angle $ heta $ on the unit circle where the equation $ cos^2( heta) - sin^2( heta) = 1 - 2sin^2( heta) $ holds true
Answer 1
To solve for $ \theta $ on the unit circle in the equation $ \cos^2(\theta) – \sin^2(\theta) = 1 – 2\sin^2(\theta) $, start by using trigonometric identities:
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We know that $ \cos^2(\theta) = 1 – \sin^2(\theta) $, so the equation becomes:
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$ 1 – \sin^2(\theta) – \sin^2(\theta) = 1 – 2\sin^2(\theta) $
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Simplify both sides:
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$ 1 – 2\sin^2(\theta) = 1 – 2\sin^2(\theta) $
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The equation holds for any $ \theta $ where $ 1 – 2\sin^2(\theta) $ is defined, which simplifies to $ \theta = n\pi $, where $ n $ is an integer.
Answer 2
Given $ cos^2( heta) – sin^2( heta) = 1 – 2sin^2( heta) $, we substitute $ cos^2( heta) = 1 – sin^2( heta) $:
$ 1 – sin^2( heta) – sin^2( heta) = 1 – 2sin^2( heta) $
This simplifies to:
$ 1 – 2sin^2( heta) = 1 – 2sin^2( heta) $
Therefore, $ heta = npi $, where $ n $ is an integer.
Answer 3
From $ cos^2( heta) – sin^2( heta) = 1 – 2sin^2( heta) $, substitute $ cos^2( heta) $:
$ 1 – sin^2( heta) – sin^2( heta) = 1 – 2sin^2( heta) $
This simplifies to:
$ heta = npi $
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