The unit circle has been divided into twelve equal arcs, corresponding to t-values of


0, 𝜋/6, 𝜋/3, 𝜋/2, 2𝜋/3, 5𝜋/6, 𝜋, 7𝜋/6, 4𝜋/3, 3𝜋/2, 5𝜋/3, 11𝜋/6, and 2𝜋


Use the (x,y) coordinates in the figure to find the value of each trigonometric function at the indicated real number, t, or state that the expression is undefined.
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sin 𝜋/6

Verified step by step guidance

1. Identify the angle \( \frac{\pi}{6} \) on the unit circle.
2. Recall that the unit circle is a circle with a radius of 1 centered at the origin (0,0) in the coordinate plane.
3. The angle \( \frac{\pi}{6} \) corresponds to 30 degrees.
4. On the unit circle, the coordinates for \( \frac{\pi}{6} \) are \( \left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).
5. The sine of an angle in the unit circle is the y-coordinate of the corresponding point, so \( \sin \frac{\pi}{6} = \frac{1}{2} \).

Key Concepts

Unit Circle

The unit circle is a circle with a radius of one centered at the origin of a coordinate plane. It is fundamental in trigonometry as it provides a geometric representation of the sine, cosine, and tangent functions. Each point on the unit circle corresponds to an angle measured in radians, where the x-coordinate represents the cosine value and the y-coordinate represents the sine value of that angle.

Trigonometric Functions

Trigonometric functions, including sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. In the context of the unit circle, the sine function gives the y-coordinate of a point on the circle, while the cosine function gives the x-coordinate. Understanding these functions is essential for evaluating trigonometric expressions at specific angles, such as sin(π/6).

Radians and Angle Measurement

Radians are a unit of angular measure used in mathematics, where one radian is the angle subtended at the center of a circle by an arc equal in length to the radius of the circle. The angles provided in the question, such as π/6, are expressed in radians, which is crucial for accurately determining the values of trigonometric functions. Familiarity with converting between degrees and radians is also important for solving trigonometric problems.