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. Recall that on the unit circle, any point corresponding to an angle \(t\) has coordinates \((\cos t, \sin t)\).
2. Identify the angle given: \(t = \frac{\pi}{6}\). This corresponds to one of the twelve equal divisions of the unit circle.
3. From the unit circle, the coordinates at \(t = \frac{\pi}{6}\) are known to be \(\left( \frac{\sqrt{3}}{2}, \frac{1}{2} \right)\).
4. Since \(\sin t\) corresponds to the \(y\)-coordinate of the point on the unit circle, \(\sin \frac{\pi}{6} = \frac{1}{2}\).
5. Therefore, the value of \(\sin \frac{\pi}{6}\) is the \(y\)-coordinate of the point on the unit circle at that angle.

Key Concepts

Unit Circle and Radian Measure

The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Angles on the unit circle are measured in radians, where 2π radians correspond to a full rotation of 360°. Each point on the unit circle corresponds to an angle t and has coordinates (cos t, sin t).

Coordinates and Trigonometric Functions

For an angle t on the unit circle, the x-coordinate represents cos t and the y-coordinate represents sin t. These coordinates allow direct evaluation of sine and cosine values for standard angles, such as π/6, without using a calculator.

Evaluating Trigonometric Functions at Specific Angles

To find the value of a trigonometric function at a given angle t, locate the corresponding point on the unit circle and use its coordinates. For example, sin(π/6) equals the y-coordinate of the point at π/6, which is 1/2. Some functions may be undefined if the denominator in their ratio form is zero.