What Does $sin$ Represent on the Unit Circle?
Answer 1
Abigail Nelson
On the unit circle, the function $\sin$ represents the y-coordinate of a point on the circle. The unit circle is defined as a circle with a radius of 1, centered at the origin of the coordinate system.
First, consider a point on the unit circle defined by an angle $\theta$, measured in radians from the positive x-axis. This point can be represented as $(\cos(\theta), \sin(\theta))$.
Because the radius of the unit circle is 1, the coordinates $(\cos(\theta), \sin(\theta))$ correspond to the horizontal and vertical distances from the origin.
Therefore,
$\sin(\theta)$
represents the vertical distance from the x-axis to the point on the unit circle. For example, if $\theta = \frac{\pi}{6}$ (30 degrees), the point on the unit circle is $(\frac{\sqrt{3}}{2}, \frac{1}{2})$. Thus, $\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}$.
Answer 2
Ava Martin
In the context of the unit circle, where the circle has a radius of 1 and is centered at the origin $(0,0)$, $sin$ represents the y-coordinate of a point on the circle with a given angle $ heta$. The angle $ heta$ is measured counterclockwise from the positive x-axis.
To locate a point on the unit circle corresponding to the angle $ heta$, we use the coordinates $(cos( heta), sin( heta))$.
Since the radius of the circle is 1, the coordinates $(cos( heta), sin( heta))$ provide the exact position of the point on the unit circle.
Thus,
$sin( heta)$
is the y-coordinate of the point. For instance, at $ heta = frac{pi}{4}$ (45 degrees), the point on the unit circle is $(frac{sqrt{2}}{2}, frac{sqrt{2}}{2})$, meaning $sinleft(frac{pi}{4}
ight) = frac{sqrt{2}}{2}$.
Answer 3
Henry Green
In the unit circle, $sin$ corresponds to the y-coordinate of a point at angle $ heta$. The unit circle is a circle with a radius of 1 centered at the origin.
For a given angle $ heta$, the coordinates of the point are $(cos( heta), sin( heta))$.
Therefore,
$sin( heta)$
is the y-coordinate of this point. For example, for $ heta = frac{pi}{3}$, $sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2}$.
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