Calculate $sin$, $cos$, and $ an$ values at specific angles on the unit circle

Given the angle $ \theta = \frac{2\pi}{3} $ radians, calculate $ \sin(\theta) $, $ \cos(\theta) $, and $ \tan(\theta) $.

Solution:

First convert the angle to degrees to understand its position on the unit circle: $\theta = \frac{2\pi}{3} $ radians = $120^\circ$.

From the unit circle, for $120^\circ$:

$\sin(120^\circ) = \sin(\frac{2\pi}{3}) = \frac{\sqrt{3}}{2} $

$\cos(120^\circ) = \cos(\frac{2\pi}{3}) = -\frac{1}{2} $

$\tan(120^\circ) = \tan(\frac{2\pi}{3}) = -\sqrt{3} $

Given the angle $ heta = frac{5pi}{4} $ radians, calculate $ sin( heta) $, $ cos( heta) $, and $ an( heta) $.

Solution:

First convert the angle to degrees to understand its position on the unit circle: $ heta = frac{5pi}{4} $ radians = $225^circ$.

From the unit circle, for $225^circ$:

$sin(225^circ) = sin(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

$cos(225^circ) = cos(frac{5pi}{4}) = -frac{sqrt{2}}{2} $

$ an(225^circ) = an(frac{5pi}{4}) = 1 $

Given the angle $ heta = frac{7pi}{6} $ radians, calculate $ sin( heta) $, $ cos( heta) $, and $ an( heta) $.

Solution:

First convert the angle to degrees to understand its position on the unit circle: $ heta = frac{7pi}{6} $ radians = $210^circ$.

From the unit circle, for $210^circ$:

$sin(210^circ) = sin(frac{7pi}{6}) = -frac{1}{2} $

$cos(210^circ) = cos(frac{7pi}{6}) = -frac{sqrt{3}}{2} $

$ an(210^circ) = an(frac{7pi}{6}) = frac{1}{sqrt{3}} $

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