To understand why [tex]\(\cos 60^{\circ} = \sin 30^{\circ}\)[/tex] using the unit circle, we need to delve into the properties and relationships within the unit circle, especially for the specific angles given.
In the context of the unit circle:
1. The unit circle is a circle with a radius of 1, centered at the origin [tex]\((0,0)\)[/tex].
2. Each point [tex]\((x, y)\)[/tex] on the unit circle corresponds to coordinates derived from the angle with the x-axis.
Let's focus on [tex]\(\cos 60^{\circ}\)[/tex]:
- The angle [tex]\(60^{\circ}\)[/tex] corresponds to the point on the unit circle where the [tex]\(x\)[/tex]-coordinate is the cosine of [tex]\(60^{\circ}\)[/tex] and the [tex]\(y\)[/tex]-coordinate is the sine of [tex]\(60^{\circ}\)[/tex].
- For [tex]\(60^{\circ}\)[/tex], the coordinates are [tex]\(\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\)[/tex], making [tex]\(\cos 60^{\circ} = \frac{1}{2}\)[/tex].
Now, let's focus on [tex]\(\sin 30^{\circ}\)[/tex]:
- The angle [tex]\(30^{\circ}\)[/tex] corresponds to the point on the unit circle where the [tex]\(x\)[/tex]-coordinate is the cosine of [tex]\(30^{\circ}\)[/tex] and the [tex]\(y\)[/tex]-coordinate is the sine of [tex]\(30^{\circ}\)[/tex].
- For [tex]\(30^{\circ}\)[/tex], the coordinates are [tex]\(\left(\frac{\sqrt{3}}{2}, \frac{1}{2}\right)\)[/tex], making [tex]\(\sin 30^{\circ} = \frac{1}{2}\)[/tex].
From these observations:
- The [tex]\(y\)[/tex]-coordinate (sine) of the point at [tex]\(30^{\circ}\)[/tex] is the same as the [tex]\(x\)[/tex]-coordinate (cosine) of the point at [tex]\(60^{\circ}\)[/tex].
This validates that [tex]\(\cos 60^{\circ} = \sin 30^{\circ}\)[/tex], and this can indeed be explained through the particular property in the unit circle that the coordinates at [tex]\(30^{\circ}\)[/tex] and [tex]\(60^{\circ}\)[/tex] angles have these symmetrical relationships.
The correct explanation from the given options is:
"The side opposite a [tex]\(30^{\circ}\)[/tex] angle is the same as the side adjacent to a [tex]\(60^{\circ}\)[/tex] angle; the [tex]\(y\)[/tex] ([tex]\(\sin\)[/tex]) distance of a [tex]\(30^{\circ}\)[/tex] angle is the same as the [tex]\(x\)[/tex] ([tex]\(\cos\)[/tex]) distance of a [tex]\(60^{\circ}\)[/tex] angle."
This explanation encapsulates the symmetry found in the geometry of the unit circle and the trigonometric ratios of these specific angles.
