$Understanding ; the ; representation ; of ; sine ; on ; the ; unit ; circle$

To understand what sine represents on the unit circle, let’s begin with the definition of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of the coordinate system.

Consider a point $P(x, y)$ on the unit circle that forms an angle $\theta$ with the positive x-axis. The coordinates of point $P$ can be expressed in terms of trigonometric functions as:

$x = \cos(\theta)$

$y = \sin(\theta)$

Therefore, the sine of the angle $\theta$ is the y-coordinate of the corresponding point on the unit circle.

To elaborate with a specific angle, let’s consider $\theta = \frac{\pi}{4}$. The coordinates of the point on the unit circle at this angle are:

$P\left( \cos\left(\frac{\pi}{4}\right), \sin\left(\frac{\pi}{4}\right)\right) = P\left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right)$

Thus, for $\theta = \frac{\pi}{4}$, the sine value is:

$\sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

This demonstrates that sine represents the y-coordinate of a point on the unit circle corresponding to a given angle.

To understand the representation of sine on the unit circle, we need to explore the trigonometric definition and how angles relate to coordinates.

The unit circle is a circle centered at the origin $(0,0)$ with a radius of 1. For an angle $ heta$ measured from the positive x-axis, the coordinates $(x, y)$ of the point on the unit circle are given by:

$x = cos( heta)$

$y = sin( heta)$

Thus, the sine of angle $ heta$ is the y-coordinate of the corresponding point. This means that if you move around the unit circle, the y-coordinate of the point at any angle $ heta$ will be $sin( heta)$.

For example, let’s take $ heta = frac{pi}{3}$. The coordinates of this point on the unit circle can be found as follows:

$Pleft( cosleft(frac{pi}{3}
ight), sinleft(frac{pi}{3}
ight)
ight) = Pleft( frac{1}{2}, frac{sqrt{3}}{2}
ight)$

Thus, for $ heta = frac{pi}{3}$, the sine value is:

$sinleft(frac{pi}{3}
ight) = frac{sqrt{3}}{2}$

This confirms that the sine function represents the y-coordinate of a specific point on the unit circle.

The unit circle is a circle with a radius of 1 centered at the origin. For any angle $ heta$, the coordinates of a point on the unit circle are $(cos( heta), sin( heta))$.

Therefore, the sine of angle $ heta$ is the y-coordinate of the corresponding point on the unit circle.

For instance, at $ heta = frac{pi}{6}$:

$sinleft(frac{pi}{6}
ight) = frac{1}{2}$

This shows that sine represents the y-coordinate of a point on the unit circle at angle $ heta$.

Start Using PopAi Today