60° is an angle measurement that represents one-sixth of a full circle, or one-sixth of 360°. It is a fundamental angle in trigonometry and is commonly used in the context of the unit circle.
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In the unit circle, the coordinates of the point at 60° are (\frac{\sqrt{3}}{2}, \frac{1}{2}).
The sine of 60° is \frac{\sqrt{3}}{2}, and the cosine of 60° is \frac{1}{2}.
The tangent of 60° is \sqrt{3}.
60° is a special angle in the unit circle because it is one of the angles that forms an equilateral triangle with the x-axis and y-axis.
The angle of 60° is often used in the context of regular polygons, as it is the interior angle of an equilateral triangle and the central angle of a regular hexagon.
Review Questions
Explain the significance of the angle 60° in the context of the unit circle.
The angle of 60° is significant in the unit circle because it represents one-sixth of a full circle, or 60 degrees out of the total 360 degrees. This angle is particularly important because it is one of the special angles that forms an equilateral triangle with the x-axis and y-axis. Additionally, the coordinates of the point at 60° on the unit circle are (\frac{\sqrt{3}}{2}, \frac{1}{2}), and the trigonometric functions sine, cosine, and tangent have specific values at this angle.
Describe how the angle 60° is used in the context of regular polygons.
The angle of 60° is commonly used in the context of regular polygons, such as equilateral triangles and regular hexagons. In an equilateral triangle, each interior angle is 60°, as the sum of the interior angles of a triangle is 180°, and an equilateral triangle has all sides of equal length. Additionally, in a regular hexagon, the central angle (the angle between the center of the hexagon and two adjacent vertices) is 60°, as a regular hexagon has 6 equal sides and 6 equal interior angles.
Analyze the relationship between the trigonometric functions and the angle 60° in the unit circle.
The angle of 60° in the unit circle has specific values for the trigonometric functions sine, cosine, and tangent. The sine of 60° is \frac{\sqrt{3}}{2}, the cosine of 60° is \frac{1}{2}, and the tangent of 60° is \sqrt{3}. These values are significant because they represent the ratios of the sides of a right triangle with one angle of 60°, which is a special angle in the unit circle. Understanding the relationships between the trigonometric functions and the angle 60° is crucial for solving problems involving the unit circle and applying trigonometric concepts.