Question

which of the following explains why \\(\cos 60^\circ = \sin 30^\circ\\) using the unit circle?
\\(\bigcirc\\) the side opposite a \\(30^\circ\\) angle is the same as the side adjacent to a \\(60^\circ\\) angle in a right triangle. on a unit circle, the \\(y\\) (\\(\sin\\)) distance of a \\(30^\circ\\) angle is the same as the \\(x\\) (\\(\cos\\)) distance of a \\(60^\circ\\) angle.
\\(\bigcirc\\) the side opposite a \\(30^\circ\\) angle is the same as the side adjacent to a \\(60^\circ\\) angle in a right triangle. on a unit circle, the \\(x\\) (\\(\sin\\)) distance of a \\(30^\circ\\) angle is the same as the \\(y\\) (\\(\cos\\)) distance of a \\(60^\circ\\) angle.
\\(\bigcirc\\) the ratios describe different sides of the same right triangle. on a unit circle, the \\(y\\) (\\(\sin\\)) distance of a \\(30^\circ\\) angle is the same as the \\(x\\) (\\(\cos\\)) distance of a \\(60^\circ\\) angle.
\\(\bigcirc\\) the ratios describe different sides of the same right triangle. on a unit circle, the \\(x\\) (\\(\sin\\)) distance of a \\(30^\circ\\) angle is the same as the \\(y\\) (\\(\cos\\)) distance of a \\(60^\circ\\) angle.

Explanation:

1. Recall the definitions on the unit circle: For an angle \(\theta\) in standard position, \(\cos\theta\) is the \(x\)-coordinate and \(\sin\theta\) is the \(y\)-coordinate of the point where the terminal side intersects the unit circle.
2. In a right - triangle, for complementary angles (angles that add up to \(90^{\circ}\), like \(30^{\circ}\) and \(60^{\circ}\)), the side opposite a \(30^{\circ}\) angle is the same as the side adjacent to a \(60^{\circ}\) angle. Also, the ratios \(\sin\) and \(\cos\) for complementary angles are related such that \(\sin\alpha=\cos(90^{\circ}-\alpha)\). Here, \(\sin30^{\circ}=\cos(90 - 30)^{\circ}=\cos60^{\circ}\).
3. Analyze the options:

• Option 1: It correctly states that the side opposite a \(30^{\circ}\) angle is the same as the side adjacent to a \(60^{\circ}\) angle in a right triangle and that on the unit circle, the \(y\) (sin) distance of a \(30^{\circ}\) angle is the same as the \(x\) (cos) distance of a \(60^{\circ}\) angle.
• Option 2: It incorrectly states that \(x\) is the sin distance and \(y\) is the cos distance. On the unit circle, \(\sin\theta\) corresponds to the \(y\)-coordinate and \(\cos\theta\) corresponds to the \(x\)-coordinate.
• Option 3: The statement "The ratios describe different sides of the same right triangle" is not as accurate as the description about complementary angles and the relationship between the opposite and adjacent sides for complementary angles. The first part of option 1 about the sides of the right - triangle is more precise.
• Option 4: It has the same error as option 2 regarding the correspondence of \(x,y\) with \(\sin\) and \(\cos\), and also the description of the ratios is not accurate.

Answer:

The side opposite a \(30^{\circ}\) angle is the same as the side adjacent to a \(60^{\circ}\) angle in a right triangle. On a unit circle, the \(y\) (sin) distance of a \(30^{\circ}\) angle is the same as the \(x\) (cos) distance of a \(60^{\circ}\) angle.