Find the values of $ cos(x) = -frac{1}{2} $ on the unit circle
Answer 1
Samuel Scott
To find the values of $\cos(x) = -\frac{1}{2}$ on the unit circle, we start by considering the unit circle where $\cos(\theta)$ is the x-coordinate of the point corresponding to the angle $\theta$:
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$ \cos(x) = -\frac{1}{2} $
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We know from trigonometric identities and the unit circle that:
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$ \cos(120^\circ) = \cos\left( \frac{2\pi}{3} \right) = -\frac{1}{2} $
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$ \cos(240^\circ) = \cos\left( \frac{4\pi}{3} \right) = -\frac{1}{2} $
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Therefore, the solutions in degrees are $120^\circ$ and $240^\circ$, and in radians they are:
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$ x = \frac{2\pi}{3} + 2k\pi $
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$ x = \frac{4\pi}{3} + 2k\pi \quad \text{where } k \in \mathbb{Z} $
Answer 2
Henry Green
To find the values where $cos(x) = -frac{1}{2}$ on the unit circle, use the fact that $cos( heta)$ represents the x-coordinate:
$cos(x) = -frac{1}{2}$
From the unit circle, we know:
$cosleft(frac{2pi}{3}
ight) = -frac{1}{2}$
$cosleft(frac{4pi}{3}
ight) = -frac{1}{2}$
Thus, the solutions are:
$x = frac{2pi}{3} + 2kpi$
$x = frac{4pi}{3} + 2kpi$
where $k$ is any integer.
Answer 3
Mia Harris
To find all $x$ such that $cos(x) = -frac{1}{2}$ on the unit circle, we use:
$cos(x) = -frac{1}{2}$
The solutions are:
$x = frac{2pi}{3} + 2kpi$
$x = frac{4pi}{3} + 2kpi$
where $k in mathbb{Z}$.
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