Find the angle on the unit circle in the complex plane where the cosine value is $frac{1}{2}$
Answer 1
Henry Green
We start by knowing that the cosine function gives the real part of the point on the unit circle corresponding to a given angle.
We are given $\cos(\theta) = \frac{1}{2}$ and need to find the angles $\theta$ where this holds true.
On the unit circle, $\cos(\theta)$ reaches $\frac{1}{2}$ at two points: $\theta = \frac{\pi}{3}$ and $\theta = \frac{5\pi}{3}$.
Therefore, the angles are:
$\theta = \frac{\pi}{3}, \frac{5\pi}{3}$
Answer 2
John Anderson
The cosine function represents the real part of complex numbers on the unit circle.
We have $cos( heta) = frac{1}{2}$, and we need to find the values of $ heta$ that satisfy this equation.
We know from trigonometry that $cos( heta) = frac{1}{2}$ at the angles:
$ heta = frac{pi}{3} + 2kpi quad ext{and} quad heta = -frac{pi}{3} + 2kpi$
where $k$ is any integer. For the interval $[0, 2pi)$, this simplifies to:
$ heta = frac{pi}{3}, frac{5pi}{3}$
Answer 3
Samuel Scott
The cosine function reflects the real part on the unit circle.
Given: $cos( heta) = frac{1}{2}$
Find $ heta$:
$ heta = pmfrac{pi}{3} + 2kpi$
Restricting to $[0, 2pi)$:
$ heta = frac{pi}{3}, frac{5pi}{3}$
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