Determine the trigonometric identity of $ sin( heta) $ using the unit circle
Answer 1
James Taylor
To determine the trigonometric identity of $ \sin(\theta) $ using the unit circle, we start by understanding the unit circle definition:
The unit circle is a circle with a radius of $1$ centered at the origin $(0, 0)$.
For any angle $\theta$ measured from the positive x-axis, the coordinates of the point where the terminal side of $\theta$ intersects the unit circle are given by $(\cos(\theta), \sin(\theta))$.
Therefore, the identity for $\sin(\theta)$ is the y-coordinate of this intersection point:
$ \sin(\theta) = y $
Where $y$ is the y-coordinate of the intersection point.
To provide a concrete example, if $\theta = \frac{\pi}{4}$, the coordinates of the intersection point are $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, so:
$ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $
Answer 2
Amelia Mitchell
To determine the trigonometric identity of $ sin( heta) $ using the unit circle, we know that the unit circle has a radius of $1$ and is centered at the origin.
For an angle $ heta$, the coordinates of the point on the unit circle are $(cos( heta), sin( heta))$.
Thus, the identity for $sin( heta)$ is:
$ sin( heta) = y $
For instance, for $ heta = frac{pi}{6}$, the point is $left(frac{sqrt{3}}{2}, frac{1}{2}
ight)$, so:
$ sinleft(frac{pi}{6}
ight) = frac{1}{2} $
Answer 3
John Anderson
Using the unit circle, the identity for $ sin( heta) $ is the y-coordinate of the point at angle $ heta$:
$ sin( heta) = y $
For example, if $ heta = frac{pi}{3}$, then:
$ sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2} $
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